Until a very few years since, the greater number of natural philosophers, and almost all mathematical opticians, were agreed in considering the rays of light as composed of extremely minute molecules, darted, in every possible direction, by a luminous body, with very great velocities. The form of these molecules remained undetermined; but as it was inferred from some observations, made long ago, that certain rays had not the same properties in every part of their circumference, it was natural to compare them to little magnets, and to suppose them possessed of poles. Hence the appellation of a polarised ray was applied to every ray of light so modified as to exhibit the polar properties of its molecules. We shall begin this article by describing the different methods that have been discovered for obtaining polarised rays.
SECT. I.—OF DOUBLE REFRACTION, CONSIDERED AS A MODE OF POLARISING LIGHT.
Supposing a pencil of natural light, that is to say, of light coming directly from a luminous body, without having undergone any change by refraction or by reflection, to fall upon a crystal of carbonate of lime, perpendicularly to one of its surfaces, either natural or artificial: This pencil will in general undergo a bifurcation as it enters the crystal; one half of the incident light continuing its course in a right line, according to the ordinary laws of refraction, and the other exhibiting a very remarkable phenomenon, and assuming an oblique direction, notwithstanding its original perpendicularity. The former half is called the ordinary pencil or ray, the latter the extraordinary pencil or ray. The plane passing through these two pencils must obviously be perpendicular to the surface of the crystal: this plane is of great importance in the phenomena of polarisation, and it is denominated the principal section of the crystal.
According to this definition, it appears that every ray of light, at each point of incidence on a given crystal, will have a principal section corresponding to it; and we have only to observe, that all these sections will be parallel to each other. In a rhomboid of Iceland spar, the principal section cuts the natural faces of the crystal in a line parallel to the diagonal joining the obtuse angles of the parallelogram, and dividing it into two equal parts.
Now, the ordinary and extraordinary rays both acquire within the crystal new properties, not inherent in the direct light. In order to show this, it will be sufficient to remark what happens to these rays when they fall on a second crystal which has the properties of double refraction; and, first, with respect to the ordinary ray, which has preserved its rectilinear direction without deviation.
If the principal section of the second crystal is parallel to the principal section of the first, the ray will undergo no double refraction in the second, but will continue its course without any subdivision. But if the principal sections of the two crystals are perpendicular to each other, the ray which was ordinarily refracted in the first crystal, will become an extraordinary ray in the second, and will be refracted obliquely when its incidence is perpendicular. But when the principal sections are neither perpendicular nor parallel to each other, this same ordinary ray will be subdivided in the second crystal, but its portions will be of different intensities, excepting when the sections form an angle of 45° with each other.
The extraordinary pencil exhibits phenomena of a similar nature. It remains an extraordinary pencil in every crystal of which the principal section is parallel to that of the crystal which has transmitted it, and becomes an ordinary pencil when the principal sections are at right angles to each other. It is subdivided into two pencils of equal intensity if the two sections make an angle of 45° with each other, and, in every other position, into two pencils of unequal intensities.
It is extremely easy to verify these propositions by experiment. We place on a horizontal table a piece of black paper, and draw two very fine lines at right angles to each other, with a white spot of a certain magnitude at their intersection; and we lay on the paper a rhomboid of calcareous spar, and with our eye immediately above the spot we see two images of it, situated in a line parallel to the shorter diagonal of the upper face of the rhomboid. One of these images is seen in the true place of the spot, as is easily ascertained by two lines which cross in it, and of which the portions not covered by the crystal point at it; these rays have therefore undergone no deviation, and have been ordinarily refracted. The other rays have been bent, since they do not pass through the true place of the spot, and therefore afford the extraordinary image.
We now place a second crystal on the first, in such a manner that the shorter diagonals of the faces in contact may be parallel; and we still find only two images of the white spot, but they are more remote from each other. The one retains its natural situation; whence it follows, that the rays which afford it are no more deflected from their course in the second crystal than they are in the first, and that they have always remained ordinary rays. With regard to the second image, since it is more remote from the true place of the object than when seen through the first crystal only, it is obvious that the rays, which have already been extraordinarily refracted in the lower rhomboid, have undergone the same kind of refraction in passing through the upper.
If we only turn the upper rhomboid round the vertical line, so as to remove its principal section from its parallel position, each of the two images will be subdivided into two others. The new images will at first be very weak; but their intensity will augment by degrees at the expense of the original images, in proportion as the angle formed by the two sections becomes greater; and at last, when it is a right angle, the two primitive images will have disappeared, and the new images alone will remain. One of them will be at a distance from the true place of the spot, in the direction of the shorter diagonal of the upper rhomboid, about equal to the result of the double refraction of the second crystal if it were separate from the first. This image is therefore formed by rays which have been refracted ordinarily in the first crystal, and extraordinarily in the second. And it will be equally manifest that the other image is derived, on the contrary, from rays refracted at first extraordinarily, and then ordinarily, in the respective crystals.
As it is difficult to procure very thick rhomboids of carbonate of lime that are quite transparent, the same experiments may be performed by means of two prisms, cut out... of doubly refracting crystals, and rendered achromatic by combining them with prisms of common glass placed with their bases in opposite directions. Through two such achromatic prisms placed on each other, the image of a candle appears quadruple or double, according to the relative positions of the principal sections. We thus see very distinctly that the images which disappear are not confounded with the other two, but that they become fainter gradually, whilst the others increase in intensity by the same degrees.
It appears, then, first, that the direct light is always divided into two pencils in its passage through the natural faces of a crystal of carbonate of lime; and, on the contrary, that the light of which either of the two pencils is composed, when submitted to the action of such a crystal, in some particular positions of the principal section, is not divided, and gives but a single pencil.
Secondly, the two images furnished by the direct light have always an equal degree of brightness; but the light of the ordinary or extraordinary pencils, when it undergoes a further double refraction, gives almost always images of unequal intensities.
Hence it follows, that in the act of double refraction this last light has received some new properties, by which it may always be distinguished from natural light. But are these properties necessarily of such a nature as to be inexplicable without supposing the elementary molecules of the rays to possess certain poles? This is a question which we are now to examine.
We will suppose that a rhomboid of carbonate of lime is placed horizontally; that the incident light falls perpendicularly on its upper surface, and that the principal section, which will be vertical, is in the plane of the meridian, or that it runs north and south; observing, however, that these directions are only chosen to facilitate our comprehension of the facts.
The ordinary pencil afforded by this crystal when it is submitted to the action of a new rhomboid similarly placed, that is to say, having also its principal section in the plane of the meridian, passes through it, as we have seen, without lateral refraction, and continues its course in a right line, remaining as an ordinary ray.
But when the principal section of the second crystal, being still vertical, is directed from east to west, the ordinary ray, transmitted by the first crystal, will be refracted laterally in it, although it falls perpendicularly on the surface, and will become an extraordinary ray.
In the first case, the principal section of the second rhomboid intersected the ray, or the luminous molecules supposed to compose it, from north to south; in the second, these molecules were intersected from east to west. It may be remarked, that this is the only circumstance in which the cases differ from each other, the ray falling in both cases on the same point of the crystal, and in the same angular direction. It must therefore be concluded, that in the ray of light, or in the elements of which it is formed, the north and south sides must have different properties from those of the east and west.
When we analyse the extraordinary pencil with the second crystal, if the principal section intersects this pencil from north to south, it undergoes the extraordinary refraction, but it follows the ordinary course when this same plane intersects it from east to west, which is exactly the contrary of what occurs with the ordinary pencil. The north and south sides of this pencil have therefore the properties of the east and west sides of the extraordinary pencil, and the reverse; nor is there any other difference between the pencils; the sides possessed of similar properties are only differently directed, so that if we could cause an extraordinary ray transmitted by any crystal to make a quarter of a revolution on itself, it would be impossible to distinguish it from the ordinary ray that has been separated from it.
When natural philosophers say of a loadstone or a magnet that it has poles, they merely understand by this expression that certain points about the circumference of the magnet possess properties which do not belong, either at all, or in the same degree, to the other parts of the circumference. It was then equally correct to say of the ordinary and extraordinary rays derived from the subdivision of natural light in the crystals of carbonate of lime, that they had poles, or were polarised. It is only necessary to remark, in order to avoid extending the analogy between the rays of light and the magnet beyond its proper bounds, that for every element of the ray, the sides or poles diametrically opposite to each other, that is, in the position here supposed for illustration, the north and south poles of the ordinary ray, appear both to possess exactly the same properties; and it is at the angular distance of 90° from these points, that is, on a perpendicular to the line that joins them, that we find in the same ray poles possessed of different properties; and if we compare with each other the two pencils transmitted by a given crystal, the poles possessed of the same properties will be situated in directions perpendicular to each other.
Let us once more consider the two rays transmitted by a crystal of which the principal section is supposed to be in the plane of the meridian. There is no reason whatever for assigning the denomination of poles to the north and south sides of the ordinary ray rather than to the east and west; but as it is necessary to make some distinction, it has been generally agreed to apply the name of poles to the north and south sides. Hence it has been usual to say, that the ordinary ray is polarised in the plane of the principal section; which is as much as to say that the different elements of the ray have the faces, which we have called poles, situated in that plane. The extraordinary ray is polarised perpendicularly to the principal section; its poles are situated perpendicularly to that plane, since it becomes perfectly similar to the ordinary ray when it is made to describe a fourth of a revolution round itself.
When we have arrived thus far, it becomes natural to ask whether we are to suppose that the separation of the light within the crystal has given poles to the molecules, or that the poles, already pre-existing, have merely been turned towards the same points of space. This question is a very difficult one; but we shall find hereafter, if not a demonstration of the second hypothesis, at least some plausible reason for adopting it. It will be here sufficient to remark, that the modification undergone by the rays is entirely independent of the nature of the crystal, provided that it only produce a double image; and that the phenomena presented by two rhomboids of calcareous spar, placed on each other, would be reproduced, with their minutest details, if we combined, for example, one of these rhomboids with a crystal of carbonate of lead; or if the first crystal were of sulphur, and the second of quartz or of sulphate of baryta.
But it is not only in the phenomena of double refraction that the particular properties of polarised rays are exhibited; the reflection of these rays, at the surfaces of transparent bodies, affords also a method of distinguishing them from common light.
When a pencil of natural light falls on a transparent mirror, with any inclination whatever, it is divided into two parts; the one passes through the substance of the mirror, the other is reflected. This latter portion is always found in the plane passing through the primitive pencil, and the line perpendicular to the surface, which is called the plane of reflection, and which must be carefully distinguished from the reflecting surface.
If we now place the principal section of a doubly-refracting crystal in a vertical position, and throw a pencil of common light perpendicularly on its surface, receiving the two emerging pencils on the horizontal surface of some water; and let us suppose the ordinary pencil to make with the surface of the liquid an angle of $37^\circ 15'$ (the crystal being held in a position inclined to the horizon); this pencil will undergo a partial reflection like the direct light; while the extraordinary pencil, when its angle of incidence is also $37^\circ 15'$, will enter the liquid completely, without the reflection of any of its molecules; a character which constitutes a marked distinction between this pencil and the natural light.
All other circumstances of the experiment remaining the same, let us now cause the crystal to make one fourth of a revolution round the incident pencil, so as to bring the principal section into a position perpendicular to its first situation; we shall then find that the ordinary pencil alone will be entirely transmitted by the liquid; the other will undergo a partial reflection, exactly equal to that which we had first observed in the ordinary ray; the experiment affording a new proof, that the two rays only differ in the direction which is assumed by their corresponding sides.
We find also, that in all positions of the principal section, intermediate between these two, the two pencils will both undergo a partial reflection so much the stronger, for the ordinary pencil, as the principal section is the nearer to a coincidence with the plane of reflection, and for the extraordinary pencil, as these planes are more nearly perpendicular to each other.
We shall finish this section with an account of the mathematical law which appears to determine the comparative intensities of the ordinary and extraordinary pencils into which polarised light is separated when it is analysed with a doubly refracting crystal. Let $F_o$ be the intensity of the ordinary pencil transmitted by any crystal, and $F_e$ and $F_{ee}$ the intensities of the ordinary and extraordinary pencil derived from it in passing through the second crystal: let $i$ be the angle formed by the two principal sections: then we shall have $F_o = F \cos^2 i$, $F_e = F \sin^2 i$.
In particular cases we shall have from these formulas, if $i = 0$, $F_o = F$, $F_e = 0$; if $i = 90^\circ$, $F_o = 0$, $F_e = F$; and if $i = 45^\circ$, $F_o = \frac{1}{2} F$, $F_e = \frac{1}{2} F$.
These three consequences of the formula, as we have seen, are conformable to observation. It will not, however, be proved to be mathematically exact, until we have also verified it for some values of $i$, intermediate between these limits.
The formulas belonging to the extraordinary ray are equally simple: $F_e$ being the intensity of this ray, and $F_o$ and $F_{ee}$ those of the two ordinary and extraordinary pencils into which it is divided by the crystal, preserving its former signification, we shall have $F_o = F \cos^2 i$, and $F_e = \cos^2 i$. If $i = 0$, $F_o = 0$, and $F_e = F$; and in fact there is no ordinary ray in this case, the whole light following the extraordinary path: if $i = 90^\circ$, $F_o = F_e$ and $F_{ee} = 0$; which is again confirmed by observation, since the extraordinary ray, coming from a certain crystal, follows only the ordinary course in passing through another, of which the principal section is perpendicular to that of the first. The same agreement between the calculation and the experiment will be found when $i = 45^\circ$; which, however, does not supersede the necessity of verifying these formulas, as well as the former, by direct experiments at intermediate angles.
SECTION II.—OF REFLECTION, CONSIDERED AS A MODE OF POLARISING LIGHT.
Reflection, at the surface of a transparent body, affords, as we have seen, a criterion for the distinction of polarised from ordinary rays: we must also add, that such a reflection is also capable of polarising ordinary light.
Throwing, for example, a pencil of natural rays on a mirror of common glass in a horizontal position, in such a manner that the inclination of the ray to the surface may be about $35^\circ$; we shall find that a part of the pencil will pass through the glass, the other part will be reflected, and the reflected portion will be polarised in the same manner as the ordinary pencil transmitted by a crystal of which the principal section coincides with the plane of reflection.
In fact, if we analyse the light thus partially reflected, by the assistance of a crystal of which the principal section coincides with the plane of reflection, it is not subdivided, and affords only a single ordinary image. Nor is it again subdivided in passing through the crystal when the principal section is perpendicular to the plane of reflection; but in this case it only affords an extraordinary image. In every other position we have both an extraordinary and an ordinary image, the intensity of the latter being expressed by the formula, $F \cos^2 i$; in which $F$ is the total intensity of the pencil subjected to the experiment, and $i$ the angle formed by the principal section of the crystal with the plane of reflection. This formula obviously coincides with that which we have given for the ordinary pencil in the case of two crystals combined. The plane of reflection here performs the office of the principal section of the first crystal; it is therefore in this plane that the ray has become polarised by the reflection.
Before we assert, however, that there is an identity in the species of polarisation effected by partial reflection with given inclinations, at the surface of a transparent body, and that which results from double refraction, we must submit the ray polarised by a first reflection to the test of new reflections.
These second reflections will obviously throw the light downwards if the mirror is above the ray, or upwards if below; from right to left if the mirror receives the ray on its [left-hand] surface, and the reverse if on the [right] surface.
Now, if the second mirror is above or below the ray, so that the new plane of reflection coincides with the old, there is a partial reflection at all incidences. But when, on the contrary, this mirror is presented to the ray on the left or the right side, and in such a manner that the new plane of reflection may be perpendicular to the old one, all reflection ceases at the inclination of about $35^\circ$, already mentioned. In the intermediate positions of the mirror, and with the constant inclination of $35^\circ$, the intensity of the reflection varies in proportion to the square of the cosine of the angle formed by the two planes of reflection with each other.
The least attention will show how much analogy this experiment has with those which have been made with a rhomboidal crystal. In those experiments, in order to see if the ordinary ray had the same properties with respect to each point of its circumference, we caused the crystal to revolve round the ray as an axis, so as to bring in succession its principal section, and consequently the poles contained in it, into a vertical direction, from right to left, and so forth; in these different positions we threw the ray on a transparent substance. Here we left the first plane of reflection immovable, and the second turned round the
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1 See the article Optics, p. 462. Polarisation of Light.
Ray, which was thrown on its different sides. This trial is evidently similar to the former, and the result is identical.
We may therefore now affirm, that the ray which is reflected at the upper surface of the glass, with an inclination of about 35°, possesses in all respects the same properties as a ray transmitted by a crystal of which the principal section coincides with the plane of reflection.
We have supposed the substance employed in this experiment to be transparent; but we must add, that some opaque bodies, such as black marble, ebony, and some varnishes, possess in an equal degree the property of polarising rays which are reflected at their surface. These substances, when made to revolve round a polarised ray, exhibit the same effect, with regard to the reflected light, as the transparent substances which have been considered.
Sect. III.—Of Rays Partially Polarised.
Rays partially polarised are those which possess properties that may be called intermediate between the properties of ordinary light and of light completely polarised. They are distinguished from polarised light by affording always two pencils in their passage through a crystal possessed of the property of double refraction; they differ from natural light in not affording always two pencils of the same intensity, in all positions of the principal section of the same crystal.
It may be asked, if a ray partially polarised may not be considered as consisting of a portion A of polarised and a portion B of natural light. This latter portion would always be equally divided into an ordinary and an extraordinary pencil, in its passage through a crystal possessing the properties of double refraction; the other would pass sometimes entirely as an ordinary or an extraordinary pencil. In a certain position of the principal section, therefore, the comparative intensities of the two pencils would be \( \frac{1}{2}B + A \) and \( \frac{1}{2}B + A \); and, after one fourth of a revolution, they would become \( \frac{1}{2}B + A \) and \( \frac{1}{2}B + A \) respectively. In all other positions, A would be divided between the two images; the portion belonging to the ordinary image being expressed by \( A \cos^2 i \), i being the angle contained by the plane of polarisation of A and the principal section of the crystal; and when \( i = 45^\circ \), the two images would be of equal intensity.
All these consequences of the hypothesis that we have assumed are conformable to experiment; and we may therefore suppose that a ray partially polarised is composed of two separate portions, the one, B, in its natural state, the other, A, totally polarised. (Sir David Brewster's experiments on partial polarisation will be mentioned hereafter.)
In every pencil reflected perpendicularly by a transparent substance, the portion A vanishes: it acquires greater and greater values in proportion as the angle of incidence increases; but at the angular situation of complete polarisation B vanishes, and A comprehends the whole pencil. Still farther from the perpendicular, we find again in the pencil natural light B, and polarised light A. Lastly, when the incident and reflected light sweeps, as it were, the surface of the mirror, A is again very inconsiderable with respect to B.
Metallic mirrors do not completely polarise the rays that they reflect at any angle of incidence. As in the case of transparent substances, A is evanescent for perpendicular rays; but it becomes sensible in every other case, and the light becomes partially polarised. The angle of polarisation of a metal is that which makes the quotient \( \frac{B}{A} \) a maximum.
There exist also some transparent bodies, such as the diamond and sulphur, which never produce complete polarisation of the light that is reflected at their surfaces; but the quotient \( \frac{A}{B} \) acquires, at least, much greater values for their substances than for the metals.
The mathematical law which connects the value of A with that of the angle of incidence, and of the refractive form of the mirror, has not yet been discovered. It is only known, that at regular angular distances above and below the angle of complete polarisation, the proportion of A to \( A + B \) is nearly the same, although the absolute value of A and B may have changed very considerably.
Thus, in the case of the glass of St Gobin, for example, in which the complete polarisation takes place when the inclination of the ray to the surface is about 35°, we find that the reflected pencils contain the same proportion of polarised light at the following angles:
| Angle | Proportion | |-------|------------| | 65° 42' | 63° 54' | | 60° 18' | 7 12 | | 7 55 | 11 40 |
For water, the relation of A to \( A + B \) is nearly the same at the angles 35° 29' and 73° 48'; the mean of these, 38° 36', exceeds by 11° only the true inclination of complete polarisation, though it is deduced from angles which differ from it more than 30°.
In the same manner, therefore, as astronomers determine the instant of the passage of a luminary over the meridian, by corresponding altitudes, observed before and after that passage, we may obtain, with tolerable precision, the angle of complete polarisation, by taking the half sum of the inclinations corresponding to equivalent partial polarisations, especially if we take care not to deviate too far from the angle required; and this method has its advantages, when we make the experiment on bodies which do not polarise the rays of light completely at any incidence.
Sect. IV.—Of the Laws Which Connect the Refractive Densities of Bodies with the Angles of Polarisation.
It is sufficient to look over the tables which have been published, of the angles of complete polarisation for rays reflected by substances of various kinds, in order to observe that these angles, reckoned from the perpendicular, approach so much the more to right angles, as the refractive densities of the substances are greater; but it was not so easy to detect the remarkable connection which exists between these two elements, and which we shall now proceed to examine.
When a ray of light, IO, passes from a vacuum into a certain medium, SS', it is refracted at the point of incidence O, approaches to the perpendicular PQ, and follows, for example, the direction OR; the angles POI and QOR being for each medium connected by the proportion \( \sin \text{POI} : \sin \text{QOR} = m : 1 \), in which the quantity m is constant for all values of the angles. This quantity, which is always greater than unity, has been called the index of refraction appropriate to the medium. It is necessary to distinguish it carefully from the refractive power, a numerical expression depending on m, and on the density,
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1 See the article Optics, p. 465, col. 2, and p. 469, col. 1. and chiefly relating to the particular properties attributed to refractive substances in the theory of emission.
This being understood, if the angle of incidence be supposed such, that the reflected ray OF may be completely polarised, it is found that the tangent of the angle of incidence will be equal to the index of refraction.
In the following table, the angles of polarisation, determined by experiment, are compared with those which result from the general law; and the errors are not greater than may be attributed to the unavoidable error of observation.
| Substances | Observed Angle | Computed Angle | Difference | |---------------------|---------------|----------------|------------| | Air | 45° to 47° | 45° | [1°] | | Water | 52 45° | 53 11° | -26° | | Fluor spar | 54 50° | 55 9 | -19° | | Obsidian | 58 3° | 56 6 | -3° | | Sulphate of lime | 55 28° | 56 45 | -17° | | Rock-crystal | 57 22° | 56 58 | +24° | | Topaz | 58 40° | 58 34 | +6° | | Iceland crystal | 58 23° | 58 51 | -28° | | Ruby spinel | 60 16° | 60 25 | -9° | | Zircon | 63 8° | 63 0 | +8° | | Glass of antimony | 64 45° | 64 30 | +15° | | Sulphur | 64 10° | 63 45 | +25° | | Diamond | 68 2° | 68 1 | +1° | | Chromate of lead | 67 42° | 68 3 | -21° |
This law is capable of being expressed in other two remarkable forms:
Since in all cases $\sin \text{POI} : \sin \text{QOR} = m : 1$, we have universally $\sin \text{POI} = m \sin \text{QOR}$; but for the angle of complete polarisation $\tan \text{POI} = m$, and since $\tan \text{POI} = \frac{\sin \text{POI}}{\cos \text{POI}} = m$, and $\sin \text{POI} = m \cos \text{POI} = m \sin \text{QOR}$, and $\sin \text{QOR} = \cos \text{POI}$, consequently $\text{QOR} + \text{POI} = 90°$; hence, when the polarisation is complete, the inclination of the incident ray to the surface is equal to the angle of refraction; and the reflected and refracted rays are perpendicular to each other.
It is of consequence to examine some objections which have been made to the accuracy of the law in question. If it were mathematically accurate, the rays of different colours, it has been observed, would not be polarised exactly at the same time, since they enter the refractive medium in different directions. Hence it would follow, that in the usual mode of investigating this angle of complete polarisation by analysing the reflected light with the assistance of a doubly refractive crystal, we ought not, in any case, to lose sight of one of the images; that when we arrived, for instance, at the angle which causes all the red light of the mixed pencil to be polarised, and to pass into the ordinary pencil, the observer ought to see an extraordinary image formed of the white wanting the red, that is, green; and the same for the other colours. Notwithstanding this, it has been added, in the greater number of cases, the crystal being properly arranged, one of the images is weakened by little and little, as we approach to the appropriate inclination, and at last disappears entirely without presenting any visible traces of colour.
It may be answered, in the first place, that there actually are substances in which this appearance of colour is manifest; and which, therefore, do not polarise the rays of different kinds at the same angle, but accord with the law of the tangent. Among others may be adduced the instance of specular iron ore, in which the phenomena is very striking; and the oil of cassia, of which the great dispersive power renders it also perfectly distinct. It may be added, too, that there is every reason to hope, that more accurate observations made with homogeneous light from different parts of the spectrum, and more precise measurements of the quantities of light that escape polarisation at inclinations approaching to that of complete polarisation, will hereafter remove the slight appearance of disagreement between experiments and a law too nearly approaching to the whole mass of the phenomena to be considered as otherwise than rigorously accurate.
The table already inserted contains the names of several mediums, such as the diamond and sulphur, which do not completely polarise light; the law of the tangent seems therefore applicable to such mediums as these, provided that we understand, by the angle of polarisation, that in which the reflected pencil contains the greatest proportion of polarised light. In this case, the observation of this angle for metals would be the more important, as their refractive density has not been hitherto determined.
The angles of greatest polarisation, measured from the perpendicular, appear to be, for mercury, $76\frac{1}{2}°$; for steel, above $71°$; hence the index of refraction for mercury and for steel ought to be $4:16$ and $2:85$. [The oxide colouring the surface of heated steel has been found to give about $2:1$ for its index, which agrees sufficiently well with this experiment.]
We have hitherto only spoken of the polarisation which takes place at the first surface of transparent surfaces entered by the light; the second surfaces possess analogous properties with respect to light passing out of them.
The angle measured from the perpendicular at which light is polarised when it is on its passage from a vacuum into a refractive medium, is greater than that in which the same phenomenon is observed when the light coming from the medium tends to pass into the vacuum; it is also shown by experiment that the sine of the former angle is to the sine of the latter as the index of refraction is to unity. We might express the same fact by saying, that at the second surface, as well as at the first, the ray completely polarised by reflection is perpendicular to the refracted ray. It follows, also, that if a medium is contained between two parallel surfaces, and if we throw a pencil of rays on the first surface, in the angle which affords complete polarisation, the transmitted portion of the pencil will also fall on the second surface in the angle which again produces complete polarisation.
Thus, if MN, RS, be the parallel surfaces, OR the incident ray, $n$ the index of refraction, and OV the refracted ray; the angle of refraction VOI' will always be equal to the angle OYZ, formed by the refracted ray with VZ, the perpendicular to the second surface.
Now, according to the assigned law, when POR is the angle of complete polarisation for the first surface, $\sin \text{POR} : \sin \text{angle of polarisation at the second surface} = n : 1$, whence $\sin \text{POR} = n \sin \text{angle of polarisation at the second surface}$. But from the law of the sines, as in the other cases, we have $\sin \text{POR} : \sin \text{VOP'} = n : 1$, therefore $\sin \text{POR} = n \sin \text{VOP}$.
See the article Optics, p. 404. Polarisation = n sin ZVO; an equation which, combined with the former, gives the angle of complete polarisation at the second surface = ZVO.
Hence it follows, that if the incident ray, being previously polarised, is such as to escape all partial reflection at its entrance into a substance terminated by parallel surfaces, it will also escape reflection upon its passage out of the substance.
If ROP has the appropriate value, OF becomes completely polarised, and VK likewise: the subsequent refraction in K makes no alteration in this ray; so that when we wish to procure polarised light by reflection from a transparent plate, there is no occasion to blacken the second surface of the plate; supposing it parallel to the first, the polarisation will be equally complete, and the reflected pencil will be brighter. It will only be proper to take care, by placing at a distance from the plate a black substance, as a piece of velvet, for example, to intercept the rays which might be transmitted by the plate from other objects beyond it.
The law from which these consequences have been deduced is only a particular case of a more general law, which may be thus enunciated.
The sine of the angle at which a pencil must be reflected at the first surface of a refractive medium, in order that it may contain a certain proportion of polarised light, is to the sine of the angle at which a reflection at the second face only would cause an equivalent polarisation in the same pencil, as the sine of incidence to the sine of refraction.
It has been found by experiment, that the sine of the angle at which a certain proportion of a natural pencil is reflected at the first surface of a transparent substance, is to the sine of the angle at which an equal proportion of the same pencil would be reflected at the second surface, if it arrived there immediately, as the sine of incidence to the sine of refraction. This law of photometry, combined with the former, leads to a very simple enunciation.
The first and second surface of a transparent substance polarise light in an equal degree, at the same angles which enable them to reflect light in an equal degree.
We have now seen in what manner light is polarised, either in passing from a vacuum into a given substance, or in returning out of the substance into the vacuum. We must next examine the laws relating to the polarisation which occurs at the surface separating two mediums possessed of unequal refractive powers.
Let \( m \) and \( m' \) be the indices of refraction for the two mediums: suppose \( m \) to be greater than \( m' \); it is proved by experiment that the tangent of the angle of complete polarisation at the common surface is equal to \( \frac{m}{m'} \) [which in fact is the index of refraction for that surface].
Hence it follows that the reflected and refracted rays are perpendicular to each other, as in other cases of complete polarisation.
**Sect. V.—Of Refraction Considered as a Mode of Polarising Light.**
It was believed for some time that the rays of light were only polarised by transparent substances in the act of reflection, and that the refracted pencil always retained the properties of ordinary light. But this opinion was erroneous. It is true, that the simple transmission of light through one or even two surfaces of any known substance is not sufficient for completely polarising a pencil of light; but at certain obliquities it occasions at least a partial polarisation; for, in examining the rays transmitted obliquely by a plate of glass with parallel surfaces, by means of a crystal possessed of the property of double refraction, it is found that the two images differ sensibly in intensity.
Let \( A' \) be the part of the transmitted pencil which is polarised, \( B' \) the portion of the same pencil not so modified, and \( A \) and \( B \) the analogous portions of the pencil reflected at the same incidence.
We have already said that \( A \) is polarised as an ordinary ray would be, transmitted by a crystal having its principal section coinciding with the plane of reflection; the poles of \( A' \), on the contrary, are placed like those of the extraordinary ray of the same crystal. If we analyse with a rhomboid the light reflected by a plate of glass, we find, for example, when in a certain position of the principal section, the image on the right hand is the more brilliant; and the relative position of the plate, the crystal, and the eye, remaining unaltered, it will be the opposite image, on the left hand, that will become the most conspicuous when we examine the transmitted light. All these results may be thus enumerated.
The plane which contains the poles of the transmitted light is perpendicular to that which contains the poles of the reflected light.
These different pencils are therefore polarised in directions at right angles to each other.
If we submit the reflected light \( A + B \) to the action of a crystal of carbonate of lime, of which the principal section coincides with the plane of reflection, the intensities of the ordinary and extraordinary pencil will become \( A + \frac{1}{2} B \) and \( \frac{1}{2} B \).
In the same position of the plate and of the crystal, the intensities of the same images, furnished by the transmitted pencil, will be \( \frac{1}{2} B' \) and \( A' + \frac{1}{2} B' \).
\( A \) will therefore be the quantity by which the ordinary image, furnished by the reflected pencil, surpasses the extraordinary image, while \( A' \) will express, for the transmitted pencil, the quantity by which the extraordinary image will surpass the ordinary. With respect to this phenomenon, a remarkable result of experiment may here be noticed; that is, that in every possible inclination \( A = A' \); or, in other terms, that the intensities of the two images afforded by the crystal differ in the same degree, whether we consider the reflected or the transmitted pencil.
Let us suppose that a plate of glass ED is placed in the position that the figure represents, before a medium AB of a uniform tint, for instance a sheet of fine white paper. The eye placed at O will receive simultaneously the ray IO, reflected at I, and the ray BIO transmitted at the same point. Place at \( mn \) an opaque diaphragm, blackened, and perforated by a small hole at S. Lastly, let the eye be furnished with a doubly refracting crystal, C, which affords two images of the aperture.
If, now, by means of a little black screen placed between B and I, we stop the ray BI, which would have been transmitted, the crystal, properly placed, will give an ordinary image \( = A + \frac{1}{2} B \), and an extraordinary image \( = \frac{1}{2} B \). But if the screen were placed between A and I, and the ray AI were intercepted, we should still have two images of the hole, and their intensities would be \( \frac{1}{2} B' \) and \( A' + \frac{1}{2} B' \).
---
1 See the article Optics, p. 470-473. respectively. Consequently, without any screen, if the whole of the reflected light AIO and the transmitted B1O are allowed to arrive at the eye, we shall have for the ordinary image, $A + \frac{1}{2}B + \frac{1}{2}B'$; and for the extraordinary, $\frac{1}{2}B + A' + \frac{1}{2}B'$.
Now it appears, from actually making the experiment, that the two images are perfectly equal, whatever may be the angle formed by the ray AI with the plate of glass, which can only be because A is always equal to A'. Consequently,
The quantity of polarised light contained in the pencil transmitted by a transparent plate, is exactly equal to the quantity of light polarised at right angles, which is found in the pencil reflected by the same plate.
Hence it follows, that at the angle of complete polarisation by reflection, the two images of the transmitted pencil given by a crystal properly placed, differ in intensity by a quantity equal to the whole of the reflected pencil. So that if ever a body should be discovered that could reflect half of the light incident at this angle, the pencil transmitted, at the same inclination, would also be completely polarised.
In order to simplify this reasoning, we have supposed throughout that there was only one efficient surface in the plate ED; it would be entering into too much detail if we proceeded to demonstrate in what manner this supposition might be justified; and it will be sufficient to remark, that the experiment in question succeeds equally well when ED is a simple plate of glass with parallel surfaces, which implies that the second surface polarises also equal quantities of light by reflection and by refraction. But, lastly, in order to remove every doubt respecting the accuracy of these results, it may be added, that when some natural light, and some light that had passed through a rhomboid of carbonate of lime, was thrown on a plate of glass reflecting at one of its surfaces, or at both, it was found that the reflected light contained the same quantity of polarised rays in both cases. Now the reflecting plate exercises no particular action on the two equal pencils of light transmitted by the rhomboid, and polarised in directions perpendicular to each other; it only divides them unequally; and if the reflected pencil contains an excess of rays polarised in one direction, there will be found in the transmitted pencil an excess precisely equal, of rays polarised in the direction perpendicular to it. In this case, the law here laid down must necessarily be true; and in order to extend it to natural light, it is sufficient to have assured ourselves, as we have done, that it is affected in the act of reflection and in that of refraction, precisely in the same manner as the combination of two equal pencils polarised at right angles to each other.
One of the best modes of verifying the accuracy of physical laws is, to inquire what their results are in extreme cases. The law now in question, supposing it universally true, leads us, for example, to this conclusion, that where there is no transmission of light, there can be no polarisation; and, in fact, if we cause a pencil of light to fall on the interior surface of a prism, at an inclination which produces complete reflection, we shall find no trace of polarisation in the reflected pencil, although, at incidences but little different, a considerable part of the light, and even the whole, may have been polarised.
Let us represent by A the part of the pencil I which is polarised by reflection, at the angle of 35° from the two surfaces of a plate of glass; the pencil transmitted will be I—A; now in this quantity of light there will be, according to the law laid down, A rays of which the plane of polarisation is perpendicular to that of the reflected rays, so that the quantity of natural light remaining will be $I - 2A$, which we may call P. Then the pencil $P + A$ falling on a second plate, parallel to the first, and consequently at the same angle as before, A will entirely escape the partial reflection, and, abating what may be lost by absorption, will be found entire in the pencil transmitted by the second plate; P will be divided in the same manner as I was at first; a portion $A'$ of P is polarised by reflection, the remaining portion $P - A'$ is transmitted, and will contain $A'$ of polarised rays, so that the natural light, after passing through the second plate, will be reduced to $P - 2A'$, and the quantity polarised by refraction will be $A + A'$. Making $P - 2A' = P$, the light $P + A + A'$ will furnish, in its passage through a third plate, parallel to the two former, a new quantity $A''$ of light polarised by refraction, which will be added to $A + A'$, and so forth.
The pencils, I, P, P', and so forth, consisting of natural light, are polarised in equal proportions by the respective plates; the quotients $\frac{A}{I}, \frac{A'}{P'}, \frac{A''}{P''}$, will constantly have the same value. If, for instance, $\frac{1}{4}$th part of the pencil I is polarised by reflection at the surfaces of the first plate, $\frac{1}{4}$th of P will be polarised at the surfaces of the second, $\frac{1}{4}$th of P by the third, and so forth; and the pencils transmitted by these same plates will contain respectively of natural light, or of light possessing the same properties, $\frac{1}{4}$ths of I, $\frac{1}{4}$ths of P, $\frac{1}{4}$ths of P', and so forth. Hence, whatever may be the number of plates employed, the pencil ultimately transmitted will contain, mathematically speaking, a certain quantity of natural light; but this quantity will be rapidly diminished, and will finally become completely insensible. [Sir David Brewster, on the contrary, maintains that it wholly disappears after transmission through a finite, and even a moderate number of plates.]
It may be said, in this sense, that a pile of parallel plates polarises the light which passes through them, in a direction perpendicular to the plane in which the rays would be polarised by reflection at the same surfaces.
We have here supposed the incident light to meet the plate of glass at the inclination capable of polarising it completely by reflection; but the same result is obtained whatever the inclination may be. It is only necessary that it should be composed of a number of elements, so much the greater as the direction of the rays is nearer to the perpendicular.
With a given inclination, the number of plates necessary to produce by transmission a polarisation nearly complete, depends also on their reflective power: we have already observed, for example, that a single plate, capable of reflecting half the incident light at the angle of polarisation, would of itself be sufficient to constitute a pile.
There are certain natural bodies, the agate, for example, which modify the transmitted light precisely as a pile of plates would do. If we cut a plate of agate sufficiently thick, in a direction perpendicular to that of its laminae, the light which passes through it acquires a polarity in the direction of the plates. A similar property is observed in the tourmaline, and it is here the more remarkable, as this mineral, when pure, exhibits no lamellar structure whatever. If the two opposite faces of a prism of tourmaline are polished so as to form it into a plate with parallel surfaces about $\frac{1}{2}$th of an inch in thickness, the light transmitted by it, whatever its angle of incidence may be, will be polarised in a direction perpendicular to the axis of the column.
It will be proper to mention here the phenomena exhibited by piles of plates when they are exposed to rays that... have been previously polarised: supposing always that the pile is formed of parallel plates of glass; and, besides, that the angle of inclination to the first surface is about $35^\circ$, reckoning from the surface itself.
If the primitive plane of polarisation of the incident ray coincides with the plane drawn through this ray and the perpendicular to the first plate at the point of incidence, a part of this ray more considerable than if we employed natural light will be reflected; at the point of incidence on the second plate, the luminous pencil transmitted by the first will undergo reflection in the same proportion as the former pencil; the same effect will take place in the third plate, and so forth. The transmitted ray, however intense it may have been in the first instance, will thus be gradually weakened in a geometrical progression, and at last will become insensible; so that if we look at the pile on the opposite end, it will appear to be an opaque body, perfectly impervious to light.
Everything else remaining in the same state, if we turn the pile round the ray as an axis through an angle of $90^\circ$, the new plane of reflection will be perpendicular to the former, and the plate will have reached the situation which has been already mentioned, in which the reflective property wholly disappears, and the whole of the light will pass through it. But the second plate, the third, and all the following plates, which are parallel to the first, will be found in the same circumstances possessing the same properties, and will not reflect any of the incident light; so that, setting aside the effect of absorption, the apparatus actually transmits light without weakening it.
The pile of plates possesses, therefore, with respect to polarised light, the singular property of being either completely opaque or perfectly transparent, according to the side which it presents to the light, notwithstanding that the inclination of the light to the first surface remains constantly $35^\circ$. In the intermediate positions the quantity of transmitted light increases gradually as we proceed from that in which nothing is transmitted, to the other extreme in which the light is only weakened by absorption.
Tourmalines and agates appear to be true piles of plates, so that they must produce similar effects; and, in fact, a plate cut, for example, in a direction parallel to the axis of a column of tourmaline, transmits rays which are polarised in a plane perpendicular to the axis, and totally stops, on the other hand, rays of which the primitive plane of polarisation is parallel to that axis.
When we place such a plate between the eye and a reflecting surface of water or glass situated in the open air, and look at it with an inclination producing complete polarisation, it appears either fully enlightened or quite dark, or in intermediate states, according to the situation of the plate in its own plane. A circumstance which adds to the singularity of this experiment is, that it succeeds completely even when the incidence on the plate is perpendicular; while for a pile properly so called, unless it be composed of an immense number of plates, it is necessary that the distance of the ray from the perpendicular should be very considerable.
Whatever may be the cause of these phenomena, it results evidently, from what has been mentioned, that two plates of tourmaline placed so that their axes form a right angle, must compose a system perfectly opaque with respect to light of all kinds. If, for example, the incident light is in its natural state, it is obvious that the portion transmitted will be polarised in the direction of the axis of the plate, and that the second plate, situated in a perpendicular direction, will consequently stop the whole of the light so polarised.
After having examined in what manner ordinary light is converted into polarised light, we must now proceed to study the modifications which this latter undergoes in its turn when it is subjected to reflection or refraction at surfaces of different natures, and differently situated with respect to its poles.
When a polarised pencil falls on the surface of a well-polished mirror, in such a manner that its plane of polarisation coincides with the plane of reflection, or is perpendicular to it, the light regularly reflected at this surface is completely polarised, like the incident pencil, in a direction parallel or perpendicular to the plane of reflection; and this happens whatever the nature of the mirror may be.
But whenever the primitive plane of polarisation of the incident pencil is any otherwise situated, it will be found that the reflected pencil is modified, and the modification will depend on the nature of the mirror.
When we employ for these experiments a mirror either transparent or opaque, which is capable of completely polarising natural light, the rays previously polarised which fall on this surface will again be completely polarised, after their reflection, but not in the plane of their primitive polarisation. This deviation of the plane of polarisation of a luminous pencil, produced by its reflection at the first surface of a transparent mirror, depends both on the angle of incidence and on the direction of the plane of reflection with regard to the poles of the ray.
For given inclination, the deviation is so much more considerable as the plane of reflection makes with the plane of primitive polarisation an angle more nearly approaching to $45^\circ$.
Let us first suppose, in order to assist the imagination, that the reflecting surface is horizontal; that the eye of the observer, and the rhomboid which is to furnish the polarised pencil, remain constantly situated, the one to the north, the other to the south of the point of reflection, so that the plane of reflection may always coincide with the meridian; and that, lastly, the principal section of the crystal, which contains in its plane the poles of the ordinary pencil, makes an angle of $45^\circ$ with the meridian.
When this ordinary pencil falls perpendicularly on the mirror, it will be reflected without any deviation of its plane of polarisation: so that this plane, having at first formed by the suppositions, an angle of $45^\circ$ with the meridian, the inclination to the meridian will remain the same after the reflection.
If we cause the direction of the incident light to vary more and more from the perpendicular, we shall first remark that the plane of polarisation of the reflected light approaches by degrees to the plane of reflection, and that it coincides exactly with it when we have arrived at the angle of complete polarisation; that afterwards the reflected ray is polarised in a plane which is more remote from the plane of reflection in proportion as it forms a smaller angle with the surface of the mirror; and that at last, when the ray is nearly parallel to the surface, its plane of polarisation coincides with that of the incident light, as it did when the incidence was perpendicular.
Let us call the angle of incidence, reckoned from the perpendicular, $\theta$, the corresponding angle of refraction for the substance concerned, $\phi$; the tangent of the angle formed by the plane of polarisation of the reflected light with the plane of reflection will be expressed by
$$\frac{\cos(\theta + \phi)}{\cos(\theta - \phi)}.$$ This formula may be illustrated by applying it to particular cases. If \( i = 0 \), \( r \) being also \( = 0 \), \( \cos (i + r) = 1 \); but the angle of which the tangent is unity is an angle of \( 45^\circ \). Consequently, if the formula is correct, the plane of polarisation for the reflected ray, when the incidence is perpendicular, must coincide with the primitive plane of polarisation of the light employed; and this is conformable to observation.
The angle of which the tangent is \( \frac{\cos (i + r)}{\cos (i - r)} \) becomes also \( 45^\circ \) when \( i = 90^\circ \); that is to say, when the rays are parallel to the surface concerned, since then \( \cos (i + r) = -\cos r \) and \( \cos (i - r) = +\cos r \). The light preserves, therefore, in this case also, its primitive plane of polarisation, as the experiment had shown.
If \( i + r = 90^\circ \), the angle of \( i \), as it has already been observed (Sect. III.), is that of complete polarisation, and \( \frac{\cos (i + r)}{\cos (i - r)} = 0 \); so that the plane of polarisation of the reflected ray coincides with the plane of reflection, which has already been shown by experiment.
The following table will show that, for intermediate angles of incidence, the agreement between this mode of calculation and the observation is as satisfactory as it was possible to expect.
### On Glass
| Angles of Incidence | Observed Deviation of the Plane of Polarisation | Computed Deviation | Difference | |---------------------|-----------------------------------------------|-------------------|-----------| | 24° | 38° 55' | 37° 54' | +1° 1' | | 39 | 24 35 | 24 38 | -0 3 | | 49 | 11 45 | 10 52 | +0 53 | | 56½ | 0 0 | 0 0 | 0 | | 60 | 5 15 | 5 29 | -0 14 | | 70 | 19 52 | 20 24 | -0 32 | | 80 | 32 45 | 33 25 | -0 40 | | 85 | 38 55 | 39 19 | -0 24 | | 87 | 40 55 | 41 36 | -0 41 | | 88 | 41 15 | 42 44 | -1 29 | | 89 | 44 35 | 43 52 | +0 43 |
### On Water
| Angles of Incidence | Observed Deviation | Computed Deviation | Difference | |---------------------|--------------------|--------------------|------------| | 53° | 0° 0' | 0° 0' | 0° 0' | | 60 | 10 20 | 10 51 | -0 31 | | 70 | 25 20 | 24 48 | -0 32 | | 80 | 36 20 | 35 49 | +0 31 | | 85 | 40 50 | 40 32 | +0 18 |
The formula, thus compared with experiment, supposes that the primitive plane of polarisation of the light employed makes an angle of \( 45^\circ \) with the plane of reflection; but a slight addition is sufficient to accommodate it to all other cases. If we make \( a \) the angle of which the particular value is here assumed \( 45^\circ \), \( i \) and \( r \) retaining their values, the tangent of the angle expressing the deviation of the plane of polarisation of the incident pencil, after reflection, will in general be represented by \( \frac{\cos (i + r)}{\cos (i - r)} \tan a \).
It is easy to observe, that, in the most material cases, this formula correctly corresponds with the experiments; but we are still in want of an experimental demonstration of its truth for any very diversified combinations of the values of \( a \) and \( r \).
The deviations of the planes of polarisation follow the same gradations when the reflection takes place at the second surface of transparent mirrors, from the position of perpendicular incidence to that of the beginning of total reflection. But beyond that inclination the phenomenon acquires a character totally different: we have then no longer a simple change of the direction of the primitive poles of the ray; for unless the plane which contains the poles either coincide with the plane of reflection or be perpendicular to it, the ray will undergo a true depolarisation, so that however we place the rhomboid through which we cause it to pass, we shall always observe that two images are formed. The same happens also when the mirror is metallic. The particular and very remarkable modifications which the light undergoes in these two cases will shortly be mentioned.
### Sect. VII.—Of the Phenomena of Interference, so far as they are modified by a Previous Polarisation of Light.
It has long been known [having, however, first occurred to the translator of this article in the room and at the table on which he is now writing], that if, after having cut in a thin plate of metal two very fine slits very near to each other, we cause them to be enlightened by a pencil proceeding from the same radiating point, we may observe behind the plate a formation of iridescent fringes, derived from the action exerted by the rays scattered from the left-hand slit on the rays scattered from the right-hand slit, in the points where these two parcels of rays are intermixed.
This experiment, when studied in all its details, has led to the simple law, which may be thus enunciated: Two rays of homogeneous light, proceeding from the same source, and arriving from a given point of space by two different routes, a little unequal in length, co-operate with each other, or are destroyed, and form, on a screen which receives them, either a bright or a dark spot, according to the magnitude of the difference of their routes.
The two rays always co-operate completely when they are united after passing through a route of equal length. If the smallest difference of routes, which will cause them again to co-operate, be called \( d \), they will co-operate wherever the distance is any member of the series \( 2d, 3d, 4d, \ldots \); and the intermediate values \( 0 + \frac{1}{2}d, d + \frac{1}{2}d, 2d + \frac{1}{2}d, \ldots \), will show the cases in which the two rays, when combined, produce darkness. The magnitude of the quantity \( d \) varies with the species of light concerned, and with the nature of the medium which transmits it.
If two rays destroy each other after having passed through routes differing, for example, by the quantity \( d \), they will also destroy each other after having passed through, either perpendicularly, or with the same obliquity, two plates of the same nature, and of the same thickness.
A difference of thickness, or of refractive density, in the two plates interposed, may produce the effect of an inequality in the routes described: the difference will give rise, in certain cases, to a displacement only of the fringes; but in others they will entirely disappear.
These laws relate to the rays of light in their natural state; if we employ polarised light, we obtain results which, independently of the numerous applications of which they are susceptible, deserve for their singularity to arrest our attention.
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1 Such experiments will be found in the article Optics, p. 465, 466; and in the same article the reader will find a new formula, confirmed by numerous experiments, exhibiting the deviations of the planes of polarisation produced by refraction.
2 See the article Optics, p. 505.
3 Ibid. p. 426 and 497. Let us first suppose, that instead of enlightening the two slits of the plate of metal with natural rays, we employ polarised light; the fringes will be formed equally in both cases.
If we then try the effect of the light polarised in one direction and transmitted by one of the slits on the light polarised in a perpendicular direction, and transmitted by the other; an arrangement which may be obtained by placing two piles properly directed in the passage of the different rays; we shall find, that when the directions are correctly perpendicular, there is no trace of fringes behind the perforated plate.
It has been remarked, that any material difference in the mediums through which the two rays pass, is sufficient to annihilate the effects of interference which would otherwise be observable. The experiment now mentioned would therefore be wholly inconclusive, if we had not previously assured ourselves that the piles, which are supposed to be of the same nature, are also exactly of the same thickness. The best mode of ascertaining this is evidently to render the two planes of polarisation parallel; if in this case we perceive fringes, and if, after having turned one of the piles one fourth of a revolution round its axis without changing their mutual inclination, we find that they disappear, we may fairly conclude that this disappearance must be attributed to the direction of the polarisation of the rays of light concerned.
This experiment would be a very difficult one to make with success if the piles had any considerable thickness; but they may be made very thin by means of plates of mica, or of bits of glass blown in a lamp; and then, by dividing them in the middle, we may obtain a pair of piles of thicknesses perfectly equal. Besides, nothing prevents our varying gradually the inclination of one of the piles, so as to compensate in this manner for the effect of a slight difference of thickness, if it exists.
But there is another mode of observation which is more convenient. We take a crystal of tourmaline cut in a direction parallel to its axis, so as to form a plate with parallel surfaces; we divide it into two parts, and apply the two portions, one to the slit on the right hand of the plate, and the other to that on the left. We then find that fringes are produced when the two axes of the fragments are parallel, and that no traces of them are left when they are perpendicular to each other; and that in changing the positions of the portions of tourmaline from one of these relations to the other, the intensity of the fringes gradually disappears.
Supposing now the piles to remain fixed to the slits in perpendicular directions, so that no fringe may be visible; and placing a third pile before the eye of the observer, in a plane that forms an angle of 45° with the planes of incidence of the two former; this last pile must reduce to a similar polarisation the rays coming from the two slits, which before they met were polarised at right angles; so that there seems to be no reason why the rays should not again interfere; and yet whatever pains we take in the experiment, we shall discover no trace of fringes. [Might there not, however, be fringes if the rays met behind the third pile rather than before it? Tr.]
It is unnecessary to remark, that a tourmaline of any kind may be substituted for the third pile, as the two former may be exchanged for the two halves of a piece of tourmaline with parallel surfaces: the result will be precisely the same.
Let us lastly suppose, in order to exhaust all possible combinations, that the plate of metal is illuminated with polarised light, and that two piles or two tourmalines are so placed as to transform the rays passing through the two slits from rays polarised in the same direction, into rays polarised in directions at right angles to each other; and that, before the interference of the rays, we bring them back to a similar polarisation, either by the assistance of a third pile, or by that of a tourmaline, as in the former experiment; the observer will then find, and no doubt with great surprise, that the rays are again susceptible of interference, or that in mixing together they produce a very visible group of iridescent fringes.
The series of experiments here related leads us to the following conclusions.
1. Two pencils converted from natural light into light polarised in the same direction, retain, after this modification, the property of interfering with each other.
2. Two pencils, which are made to pass directly from the state of natural light to that of light polarised in two perpendicular directions, are no longer capable of interference, either while they remain in this state, or after they have been restored to a similar polarisation.
3. Two pencils polarised in contrary directions do not interfere, whatever may have been the modifications that they have undergone before they arrive at this state from that of natural light; but when restored to a similar state of polarisation, they become capable of interfering, provided that, in their passage from the natural state to that of polarised light, the first planes of polarisation of the two pencils were parallel.
Thus it appears, that in these phenomena the mode of action of the rays does not depend on what they are only when they meet, but also on what they have previously been.
SECT. VIII.—OF THE KIND OF POLARISATION WHICH IS EXHIBITED IN THE APPEARANCE OF COLOURS, AND WHICH HAS THEREFORE BEEN CALLED COLOURED POLARISATION.¹
We may first examine the nature of the new modification of light that is concerned in these appearances; and, secondly, the means of producing it.
Supposing a ray of direct light to be polarised in any of the ways that have been described, and then to pass through a plate of rock-crystal cut perpendicularly to the edges of the hexadral prism, about a quarter of an inch in thickness, and having both its surfaces perpendicular to the ray: when it emerges from the surface, it will no longer possess the properties of common polarised light; and yet it will not have re-assumed the characters of direct light: for if we submit it to the action of a rhomboid of carbonate of lime, instead of affording one image only in a particular position with respect to the principal section of the crystal, it will be constantly subdivided into two pencils; so that it cannot be identical with common polarised light.
Neither is it simply natural light; for a white pencil of natural light is always divided, by a doubly refracting crystal, into two white pencils of equal intensity. The white pencil which has passed through the plate of quartz, on the contrary, gives always two images, but they are distinguished by the most vivid colours. If the ordinary image is red, the extraordinary is green, and the reverse; and the case is the same with regard to the other prismatic colours; that of the ordinary pencil is complementary to the tint of the extraordinary pencil, and they both vary according to the position of the principal section of the rhomboid which causes the separation.
The prismatic telescope of M. Rochon affords an apparatus perfectly adapted to the performance of these experiments, and which requires to be mentioned, first, because it exhibits the tints with great brilliancy; secondly, because
¹ See the article Optics, p. 476-501. it gives us the means of assuring ourselves that the images viewed lose nothing of their distinctness by the interposition of the plate, and that its effect is not owing to any irregular dissipation of the light concerned; and, thirdly, on account of the facility with which it allows us to show that the tints are complementary to each other.
This instrument is nothing more than an ordinary telescope, furnished, between the object-glass and the eye-glass, with a prism of rock-crystal or of carbonate of lime rendered achromatic. This prism being moveable at pleasure along the axis of the telescope, the observer is enabled to separate more or less, at his pleasure, the two images of the object which he is viewing.
Having, then, before the object-glass of this telescope the plate of quartz in question, and adapting besides to the eye-glass a greenish dark glass of a particular kind, which is much employed by astronomers because of its property of absorbing a good deal of light, without sensibly colouring the transmitted pencil; if we observe the sun directly with this instrument, we shall see two images of the sun, both white, and of equal brightness, whence it follows that the plate produces no particular effect on natural light; but if, on the contrary, we look at the sun's image as reflected by a plate of glass not silvered, we shall perceive two more suns, each of them coloured; and whilst the telescope performs half a revolution on its axis, they will both run through nearly the whole series of the prismatic colours. Thus, the image that was red, will become successively orange, yellow, greenish-yellow, bluish-green, and violet; and at this period the telescope will have made half a turn; its movement being continued, the violet image will become first red, then orange, and so forth. The second image will always give us the complementary colour; for if, instead of completely separating the images, we allow them to overlap each other, the part common to both the discs will remain constantly white, while the remaining lunular portions will exhibit the most vivid colours.
The reflection of a transparent plate may also be employed for more directly exhibiting the distinctive properties of the light transmitted by the plate of quartz.
If we cause a plate of glass to turn round a pencil of natural light forming with its surface an angle of about $35^\circ$, the reflected pencil will be directed in succession to all the points of the horizon, but it will constantly preserve the same intensity.
But if the incident pencil is polarised, we shall find, on the contrary, two positions diametrically opposite to each other, in which the mirror will not reflect a single ray.
Making, now, a similar experiment with the light that has been transmitted by a plate of rock-crystal, we shall see that it becomes coloured by reflection, though it falls white on the plate, and that the nature of the colour depends on the side of the ray that is presented to the reflecting surface. These reflected colours succeed each other during the revolution of the plate, in the same order as in the prismatic spectrum; and they are also observed in the transmitted light, being always complementary to those of the light reflected at the same time.
If the properties of polarised rays depend, as is supposed by the partisans of the system of emission, on the particular arrangements assumed by the molecules of which they are formed, it will be easy to describe the intimate composition of the ordinary polarised ray, and that of the same ray after it has passed through the plate of rock-crystal: in the former, the axes of all the molecules of the different colours must be parallel; in the latter, the molecules of different tints must have their poles turned towards different parts of space.
It now becomes necessary to examine according to what law the direction of these poles is varied, both as they depend on the particular tint of each molecule, and on the greater or less thickness of the crystal that they have passed through.
If we employ homogeneous polarised light, we shall readily find, that, supposing a given plate of quartz to turn the poles of a certain ray of light through an arc of $20^\circ$ from their primitive direction, a plate of the same crystal twice as thick will cause a double deviation, and will turn the pole through an arc of $40^\circ$; a plate of thrice the thickness will cause a triple deviation, amounting to $60^\circ$; and so forth, without limit.
With regard to the simple rays of different colours, in passing through a given plate, they undergo deviations so much the more considerable as they are more refrangible, and this in the inverse proportion of the numbers which Newton calls the lengths of the fits; or, what comes to the same result, in the inverse proportion of the quantities which have been designated in this article by the letter $d$. (Sect. VII.)
When, therefore, we know the deviation for a given plate, we may find the effect of a thicker or thinner plate of the same substance by a simple proportion.
Table of the Deviations of the Planes of Polarisation of the different "Homogeneous" Rays in passing through a Plate of Rock-Crystal perpendicular to the Axis of the Prism, of which the thickness is a millimetre, or $0.5937$ E. I. [according to the Newtonian division of the spectrum].
| Extreme red | 17.50° | |-------------|--------| | Limit of red and orange | 20.48° | | Orange and yellow | 22.31° | | Yellow and green | 25.68° | | Green and blue | 30.05° | | Blue and indigo | 34.57° | | Indigo and violet | 37.68° | | Extreme violet | 44.08° |
There is no reason to suppose that the angular deviations will undergo any alteration in their values when all the molecules pass through the crystal at the same time. Consequently, in the white pencil transmitted by a plate of a millimetre, the axes of the elementary red rays will form an angle of $8^\circ$ with the axes of the first orange rays, of about $5^\circ$ with those of the first yellow rays, and of $26.5^\circ$ with the axes of the extreme violet rays; and if we analyse this white pencil by means of a rhomboid, the differently coloured rays will not be divided in the same proportions between the two images: hence there will necessarily be appearances of colour. It is obvious, for example, that when the rhomboid is so placed that its principal section shall coincide with the poles of the red ray, this ray will remain altogether in the ordinary pencil, and the red tint will be wholly wanting in the extraordinary image.
We may obtain an exact idea of the modification which a plate of quartz produces in a white pencil of polarised light, by conceiving a combination of red rays polarised by reflection at the surface of a certain transparent substance, of orange rays polarised by a second surface placed in a different angular situation, of yellow rays polarised by a third surface, and so forth. The necessity of the intimate mixture of all these kinds of molecules in each line of white light, and some other obstacles, would render it impossible to realise this fiction without a very complicated apparatus; while a simple plate of quartz is sufficient, on the other hand, to give to the different constituent parts of the white pencil these individual polarisations situated in different azimuths.
The phenomena which have been described are produced by plates of quartz with parallel surfaces cut perpendicularly to the axis of the hexaedral prism. Now, in a direction perpendicular to these surfaces, quartz exhibits no double refraction; so that the causes which in this case produce the deviation of the planes of polarisation of the luminous molecules, are different from the causes which occasion the separation of the two pencils in other sections of the crystal. And it is remarkable that the properties of these plates have been found in bodies not possessed of regular crystallization, as flint-glass, and even in perfect fluids, such as the essential oils of turpentine and of lemons, or the solution of camphor in alcohol, the simple syrup of sugar, and so forth. The only difference is in the absolute value of the thicknesses which afford the same tints, the other laws remaining the same. Thus the thickness of oil of turpentine must be sixty-nine times as great as that of a plate of rock-crystal to produce the same effect; and the action of the oil of lemons is to that of the oil of turpentine as seventeen to ten [or to that of crystal as one to forty-one].
We have seen that a plate of quartz a millimetre in thickness causes the poles of the red molecules to describe an arc of 17°5°. We may suppose this motion to have taken place from right to left; then every other plate, of whatever thickness, cut out of the same crystal, will cause the poles to deviate still farther in the same direction, that is, to turn still from right to left; whilst other plates, on the contrary, cut out of a different crystal, may turn them from left to right. This phenomenon, at first sight, must appear very singular; but if we reflect that the rays pass through the plates in a direction which affords no double refraction, we shall be aware that a deviation of the poles directed constantly the same way in every specimen of the crystal would be not at all less astonishing.
It has not hitherto been possible to point out any exterior signs which shall indicate the direction of the deviation that will be produced by a given crystal, except in one very remarkable case. In some varieties of quartz, the solid angles, situated at the base of the pyramid by which the prism is terminated, are replaced by as many facets placed obliquely with respect to the edges. Now the direction of the deviation which these plagioedral crystals give to the poles of the luminous molecules, is constantly that of the obliquity of these little facets.
When a polarised ray passes successively through two plates producing contrary rotations, the ultimate deviation of the poles is the difference of the effects which each plate would have produced separately. The ray exhibits exactly the same tints as if it had passed through a single plate, of a thickness equal to the difference of the thicknesses of the two plates.
If the plates thus combined are equal in thickness, the pencil transmitted, having been turned first in one direction, and then turned back in the contrary direction, seems not to have its polarisation ultimately changed.
The essential oil of turpentine causes the axes of the molecules of the polarised ray to turn from the right to the left of the observer receiving the ray; the essential oil of lemons from left to right. These fluids do not lose their peculiar properties when they are mixed; so that if their proportions in the mixture are inversely as their rotatory forces, the ray which has passed through them retains its primitive polarisation.
For this purpose we may place the principal section of a rhomboid of calcareous spar in the plane of polarisation of a white pencil, which of course will be subjected to the ordinary refraction only; we may then place the plate in question before the rhomboid, so that the rays may pass through it perpendicularly. If now the principal section of this plate is parallel to that of the rhomboid, we shall still see but one white image; and the same will happen if the principal sections are perpendicular to each other; but in every other situation of the plate, the rhomboid will furnish two pencils, and they will be distinguished by complementary tints.
The motion of the plate in its own plane does not alter the nature of the tints; their brightness only varies, and becomes greatest when the angle formed by the two sections is equal to 45°.
These tints vary with the thickness of the plate, and degenerate into perfect whiteness when the thickness becomes considerable. In the sulphate of lime, the appearances are no longer observable when the thickness is about half a millimetre, or one fifth of an inch.
Supposing O to be the tint of the ordinary pencil, and E that of the extraordinary; the experiment shows that the tint R is nearly that of one of the coloured rings seen in the light reflected from two object-glasses touching each other, as in the celebrated experiments of Newton; and that the tint O is that of the corresponding transmitted ring. This rule, however, is not perfectly general; for in many crystals the tints E by no means resemble those of the rings.
When the regular sequence of the Newtonian rings is observable, the successive thicknesses of the same crystal, which afford the respective colours E, are proportional to those which Newton has assigned for substances not crystallized; it is only found that, for any given density, the absolute values of these thicknesses greatly surpass the thicknesses shown in the Newtonian tables.
We find also a remarkable relation between the tint E, the thickness of the plate, and the elements of its double refraction, which it is important to point out. The image E only appears when the principal section of the plate is neither parallel nor perpendicular to the primitive plane of polarisation of the ray which passes through it. If we suppose this plate to possess only the ordinary properties of double refraction, the ray will in general be divided by it into two pencils, one of which will be refracted ordinarily, the other extraordinarily; so that two pencils from the same origin meet, after having passed through different routes, and interfere. There is a certain inequality of the lengths of these paths at which the red rays destroy each other; at another interval the yellow rays, the green, the blue, and so forth. If we determine, from these principles, the tint resulting from the interference of the different rays, taking into account the thickness and the intensity of the double refraction of the plate, we shall always find a very satisfactory agreement between the calculation and the experiment. (See the article Chromatics.)
The singular deviation which these thin plates seem to produce in the poles of the molecules of different colours which constitute white light, was extremely difficult to be discovered; and nothing shows this difficulty better than the general assent of natural philosophers to the laws which have been the foundation of the theory of moveable polarisation. It is not therefore sufficient to explain here the true principles on which these phenomena are founded; the confutation of an erroneous theory becomes absolutely necessary in this stage of the investigation, especially when it is plausible in appearance, and when it is brought
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**Sect. IX.—On the Phenomena of Depolarisation, and of Colours Produced by Crystallized Plates Not Cut Perpendicularly to the Axis of Double Refraction.**
We are next to inquire how a white pencil, polarised in a single direction, is modified in passing through a crystaline plate possessed of double refraction.
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1 See the article Optics, p. 304, col. 2. 2 Ibid. p. 476. forward with confidence in the latest works, notwithstanding the decisive objections which had been opposed to it.
The fundamental theorem of this moveable polarisation has been thus enunciated. "When a ray of natural light, polarised in a certain fixed direction, passes perpendicularly through a crystallized plate, which is parallel to the axis of double refraction, the molecules of light penetrate at first to a certain distance without losing their primitive polarisation; after this they begin to perform periodical oscillations round their own centres, so that their axis of polarisation is carried alternately on each side of the principal section of the crystal, or of the line perpendicular to it, like a pendulum passing from one side to the other of the vertical line which is its quiescent position. Each of these oscillations is performed in a given thickness $2e$, twice as great as that in which the molecule had made its excursion in one direction. Thus, from the thickness zero to a certain fundamental thickness $c$, the homogeneous molecules of which the ray passing through the crystal is composed, are affected, after their emergence, as if they had not quitted their primitive polarisation; from $c$ to $2c$ they are affected as if they had assumed a new polarisation, differing from the former by the azimuth $2i$; $i$ being the angle which the principal section of the plate forms with the original plane of their polarisation; and they appear, in short, alternatively polarised in their former azimuth, and in an azimuth at the distance $2i$ from it."
To this law it may be objected, first, that whenever the light emerging from a crystal with one axis, whether thin or thick, is composed of two distinct pencils, we find that they are polarised in directions at right angles to each other, whether the incident light may have been natural or polarised; and no exception to this rule has yet been discovered. Now it is difficult, if not impossible, to conceive, in the system of moveable polarisation, how the transition can be made from that state into the state of polarisation in two directions at right angles, which, for the sake of distinction, has been called the fixed polarisation.
But there is a still more direct objection to this theory. If we place a plate of sulphate of lime in such a manner that its principal section may make an angle of $45^\circ$ with the primitive plane of polarisation of homogeneous light that is to pass through it, the angle $2i$ being then equal to $90^\circ$, the transmitted pencil, according to the doctrine of moveable polarisation, would be entirely polarised either in the primitive plane or in the plane perpendicular to it, and, when analysed by means of a rhomboid, it would exhibit, in two positions of the principal section of this crystal, only a single image. But this is so far from being true, that if the plate is of a proper thickness, the pencil will be constantly divided into two images of equal intensities, whatever the direction of the principal section of the rhomboid may be.
When two pencils, derived from the same origin, and possessing the same velocity, are made to intersect each other at a very small angle, after having travelled by different paths, of which the lengths are slightly different, they may destroy each other completely, as we have already seen; and this destruction would as readily take place if they came by the same path with different velocities. Let $d$ be the difference of the paths which determines, in the first case, the periodical series of the points of space in which two rays of a certain homogeneous light produce complete darkness by their interference; the same letter $d$ will express, in the second case, the quantity by which one of the rays will require to be more advanced than the other by the excess of its velocity, in order that the light may again be destroyed. When, in a crystallized plate, the difference of the tracks described by the ordinary and extraordinary rays of a given kind of light, or when the effect of the difference of velocity is $0$, $nd$, or $(n + \frac{1}{2})d$, $n$ being a whole number, a pencil previously polarised, and transmitted by the plate, will appear entirely polarised in the primitive plane, or in the azimuth $2i$, as the principles of the doctrine of moveable polarisation require.
When the plate has such a thickness that the difference of the paths described by the ordinary and extraordinary pencils is included in the general formula $(n + \frac{1}{2})d$, the transmitted light will appear to have become common light, if the principal section of the plate makes an angle of $45^\circ$ with the primitive plane of polarisation of the incident light, which, as we have already seen, by no means agrees with the doctrine of moveable polarisation.
Lastly, when the thickness of the plate employed is not comprehended in any of the preceding expressions, the completely polarised rays, which pass through it, emerge with the characters of a partial polarisation. This result is no less inconsistent than the former with the laws of moveable polarisation; since, according to these laws, the polarised incident ray ought always to emerge completely polarised, with a simple change of the azimuth of its poles.
It is not, therefore, generally true, that a pencil of polarised homogeneous light, which passes through thin crystallized plates, either preserves its primitive polarisation, or assumes a new one at the angular distance of $2i$; and with this fails the whole fabric of the oscillatory motions attributed to the molecules of light. With respect to the objection already made to this theory, regarding the connection to be established between the phenomena of thin plates and those of thick crystals, it seems to retain its full force, since it has not been established by experiment that the rays concerned in the phenomena of thin plates are polarised in two rectangular directions.
Supposing, however, for a moment, that this were the case, and that a luminous pencil, passing through a thin plate of sulphate of lime, is divided into two rays, the one ordinary and the other extraordinary, polarised at right angles; let us examine what would be the consequence. Mathematically speaking, these two rays follow in general different routes within the crystal; but it is not possible to separate them physically, because the imperfection of our organs forces us to contemplate objects of a certain extent. The advocates of a moveable polarisation will examine this light in a mass. They will find in certain cases that it appears to have preserved its primitive polarisation, and in others that it seems to be polarised in an azimuth differing by $2i$, and they will hence conclude that thin plates act very differently from a thick crystal.
This conclusion, however, may be disputed. When we make use of a thick crystal, the ordinary and extraordinary images are separated; we study the properties of each apart. In the case of the thin plates the observer has to do with light which is mixed and complicated. Now, who can affirm, without having made the experiment, that two rays really polarised at right angles, will not seem, in cases of interference, to have lost their primitive polarisations, and will not exhibit an intermediate polarisation, the result of the others, which might be considered as composing it?
The reader will now have conjectured, that, in order to elucidate these mysterious phenomena, it will require to be proved, first, that two pencils are actually formed in the thin plates of crystallized substances, as well as in thicker crystals, polarised at right angles to each other; and, secondly, that these pencils, when they are mixed, may exhibit the appearance of a polarisation intermediate between the two separate directions. Such, then, is the object of the very delicate experiments which are now to be related.
A pencil of homogeneous solar light being concentrated into a very small point, by the aid of a lens of a short focus, fixed in the shutter of a dark room, the pencil of diverging rays is to be received on two plates of glass slightly inclined to each other, and making an angle with the ray of about $35^\circ$; the pencils reflected by the two plates will then be completely polarised, and where they intersect each other they will form light and dark stripes; and whatever the position of the plates may be, the stripes will be polarised in the same direction as the two pencils which afford them.
If we now take a very transparent plate of sulphate of lime, and divide it in the middle, so as to have two plates of exactly the same thickness; if we fix one of the halves of this plate before the mirror, in such a situation that it may transmit that pencil only which is reflected by the first mirror, and so that the principal section may make an angle of $45^\circ$ with the primitive plane of polarisation; and then the other half of the plate in the way of the polarised light reflected by the other mirror, but with its principal section at right angles to that of the other plate, and making an angle of $45^\circ$ with the primitive plane in a contrary direction: then, if these plates act in the same manner as thick plates, however small the measure of their double refraction may be, they must divide the reflected rays which pass through them into two pencils of the same intensity, and polarised at right angles to each other; but, in the positions here assigned to them, it will evidently happen, that the plane of polarisation of the ordinary pencil, coming, for example, from the right-hand plate, will be parallel to the plane of polarisation of the extraordinary pencil of the left-hand plate; and the same will be true of the two remaining pencils of the respective plates.
This being admitted, it is easy to see what will happen at the points in which the two pencils intersect each other. The ordinary rays of the right-hand plate will at once interfere with the extraordinary rays of the left-hand plate, since they are polarised in the same direction, and will form a group of light and dark stripes; and a second group will be afforded by the extraordinary rays of the right-hand plate and the ordinary of its companion. The two groups will be the farther separated from each other as the plates are thicker, and as their double refraction is more strongly marked. In the intermediate space we find the rays of the same description furnished by the two plates; but as they have now received contrary polarities, they intersect each other without exhibiting any of the phenomena of interference, and the eye has the sensation of a uniform light only.
A fact which is no less evident than the existence of the two systems of stripes, is, that when we employ the sulphate of lime, each of the systems is completely polarised in a plane perpendicular to the principal section of the plate which is nearest to it.
Now, there is no one of the consequences of the supposition with which we set out, that is, the supposition that every plate divides polarised light passing through it into two pencils polarised at right angles, that is not fully confirmed by this experiment. The truth of the hypothesis is therefore demonstrated; for every other mode of separation or of polarisation of the rays, that for example which is deduced from the principles of moveable polarisation, would lead to phenomena totally different from those which have been described.
A little attention to the passages (Sect. VII.) containing an account of the circumstances under which polarised rays are capable of occasioning appreciable effects of interference, will be sufficient to convince the reader that the two systems of stripes which have been the subjects of these experiments, can only have been the result of the interference of the ordinary rays of one plate with the extraordinary rays of the other. But if any doubts were entertained on this subject, they might be removed in the following manner.
We might substitute for the two thin plates which have been employed, two thick crystals, for example two rhomboids of carbonate of lime, in which the double refraction would be manifest. As we might then follow separately the course of each pencil, and intercept them in turn by screens, it might thus be proved, by the most direct evidence, that, for the formation of the two groups of stripes, it is necessary and sufficient that the ordinary pencil of one of the crystals should meet the extraordinary pencil of the other, and the reverse. The direction of the polarisation of these fringes, determined by the aid of a rhomboid, would also be exactly the same as in the case of the employment of thin plates. The only remarkable difference between the two experiments would be in the degree of separation of the two groups of fringes. This distance, depending on the difference of the paths described by the ordinary and extraordinary rays, would be "greater" [or rather smaller] with the crystals than with the plates. It might even happen, if the crystals were very thick, that in order to "bring the stripes within the field of view" [or to render them visible], it would be necessary to compensate for a part of the difference of the lengths of the paths, or of the velocities, by the assistance of a plane glass, fixed in the way of one of the pencils; but in every case the results of the experiment would be equally clear and decisive.
It may also be added, in the last place, as a full answer to every objection that might be raised against this explanation of the formation of the two groups of stripes in the thin plates, that the interval which separates the two systems is always so dependent on the double refraction of the plates, that its exact numerical value may always be deduced from the elements of the double refraction, as obtained by experiments on other portions of the same substance. (See the article Chromatics.)
We must now consider how it will be possible to reconcile the experiment which has been related, and which proves the subdivision of the light into two pencils, polarised at right angles to each other, with the other fact, which seems to be opposed to it, that when the plate is of proper thickness, the whole group of polarised rays that pass through it shall appear to be polarised either in the primitive plane or in another making with it an angle of $24^\circ$.
We make, in a dark room, a very small radiant point of homogeneous light, by means of a lens, as already mentioned. We receive the diverging pencil on a plate of glass, having its posterior surface covered with a black varnish; supposing the plate to be in a vertical position, and the diverging pencil to be nearly horizontal, and to make an angle with the surface not much differing from that of complete polarisation: when this arrangement is completed, we place in the way of the reflected light a rhomboid of calcareous spar, its principal section making with the horizon, or with the plane of reflection, an angle of $45^\circ$. In this position of the rhomboid the light passing through it is divided into two pencils, the one ordinary and the other extraordinary, polarised at right angles, and of equal intensity. After having passed through the first rhomboid, the two pencils fall on another rhomboid of the same thickness, but having its principal section perpendicular to that of the former. The ordinary pencil will then be subjected to the extraordinary refraction in it, and the reverse; and the two pencils will emerge from the second rhomboid, one polarised in the plane of its principal section, and the other perpendicularly to it.
Let us now follow the course of each of the pencils. In the first place, it is evident, that on account of their divergence they will intersect each other in a space so much the Polarisation of Light.
Wider as they become more remote from the rhomboid; their points of emergence being distinct and sensibly separated, the observer may intercept, in turn, with a screen, either the ordinary or the extraordinary pencil, and enlighten at pleasure any other object, either with the one or the other separately, or with both at once.
This delicate and complicated experiment being so far advanced, let us place a glass slightly roughened in a part of the field of view common to the two pencils; marking by a very fine opening in an opaque plate covering the glass, the precise spot on which we fix our attention; and employing, as usual, a doubly-refracting crystal to analyse the different kinds of light, which, after passing through the slit in the diaphragm, depict an image within the eye.
It will now easily be observed that the ordinary ray, when it arrives alone at the aperture, wherever it may be placed, undergoes no modification, and remains polarised as it was before; and the same is true of the extraordinary ray: but if both these rays intersect each other in the slit, and enter the eye together, the phenomenon is by no means so simple, and its nature changes according to the place occupied by the slit; so that moving this slit gradually by means of a screw, we soon find the point where the light, composed of the two pencils that pass through it, seems to be wholly polarised, in the same manner as the pencil was at its first reflection from the plate of glass; further on, the plane of polarisation seems to be perpendicular to the primitive plane; and in a position intermediate between these the rays transmitted afford no material traces of polarisation at all.
This experiment, therefore, offers us the singular phenomenon of two pencils, polarised at right angles, which, after having intersected each other in the ground glass, unite within the eye, and form together, as the test of the rhomboid shows, a pencil polarised sometimes in one direction and sometimes in an opposite one, or sometimes without any sensible trace of polarisation, according to the magnitude of the difference of the paths described by the two pencils.
It is only to assist the imagination that the piece of ground glass has been supposed to be employed, for its presence is not necessary to the success of the experiment. A lens alone may be used for observing the stripes formed in the air by the interference of the luminous pencils. If, however, we merely placed ourselves with this lens before the two rhomboids, the eye would only receive a uniform and continued light; but as soon as a doubling crystal is perfectly interposed between the lens and the rhomboids, or between the lens and the eye, we shall observe two systems of dark and bright stripes, the bright stripes of one of the images corresponding always with the dark stripes of the other. The middle stripes, for example, will be bright in the ordinary image, if the principal section of the interposed crystal is parallel to the primitive polarisation of the rays on the blackened glass; and in the same case the middle stripe will be dark in the extraordinary image. The point of space occupied by the middle of the image seems therefore to transmit to the eye, through the crystal, only such light as is polarised in the primitive plane, because it affords only an ordinary image of the light; and this circumstance shows also that the effects of interference in the extraordinary image require, for their computation, the addition of \( \frac{1}{2}d \) to the difference of the paths described.
When the principal section of the crystal interposed between the eye and the rhomboid is perpendicular to the original plane of polarisation, the two kinds of pencils interchange their effects; and in this case the central stripe of the ordinary image is bright, and that of the extraordinary image is completely dark, as if the difference of the paths of the rays forming it were \( \frac{1}{2}d \).
It has been hitherto supposed that the original pencil contained only homogeneous light, so that it produced only bright and dark stripes. But if we employ white light, the stripes will be coloured; because \( d \) has different values for the rays of different tints, and these tints are precisely the same as are developed by polarised light in their passage through crystallized plates of all possible thicknesses.
A few words will now be sufficient to show the mode of action of these plates in the phenomena of the colours first described.
A polarised ray passing through a crystallized plate which possesses the power of double refraction is divided by it, mathematically speaking, into two pencils polarised at right angles; but two pencils of this description do not interfere; the plate will therefore not exhibit colours to the naked eye, whatever its thickness may be, even when it is only exposed to polarised light; and this result is confirmed by experiment.
Each of the ordinary and extraordinary pencils transmitted by the plate will again be divided into two when it passes through an achromatic prism of crystal, or a rhomboid of calcareous spar. Of these four emergent pencils, the two which follow the ordinary path will be no more separated than they were at their emergence from the plate; and the same is true of the two extraordinary pencils, so that the eye will ultimately perceive but two distinct images.
Of the two pencils which thus concur in the formation of the ordinary image, the one was ordinary in its passage through the plate, and has remained ordinary in the rhomboid placed near the eye, while the action of this rhomboid has been required for bringing the other pencil from the extraordinary to the ordinary refraction. The different kinds of rays have different velocities in crystals capable of double refraction; and an inequality of velocity gives rise to the phenomena of interference, as well as an inequality of distance described. If, in the plate employed, the difference between the velocities of the ordinary and extraordinary ray corresponds, either on account of its thickness or of the diversity of the two refractions, to a certain quantity, \( nd \), or its multiple, the kind of rays of which the interval \( nd \) determines the destruction, \( n \) being a whole number, will be wanting in the ordinary image transmitted by the rhomboid. And this effect, it must again be repeated, depends on the interference of the two pencils of which this image is really formed, and which, in the plate, possessed different velocities.
If the experiment with the two rhomboids had not taught us, that, in order to calculate the mutual actions of the luminous rays, which in passing through different crystals possessed of doubly refractive powers, have several times changed their planes of polarisation, the ordinary laws of interference require some modification, we should have found ourselves arrested by a considerable difficulty.
The difference of the velocities being the same for the two pencils of which the extraordinary image afforded by the rhomboid consists, and for the two which concur in the formation of the ordinary image, it would seem that the rays of the same colour ought to be destroyed at once in both the images, and that they ought to exhibit the same tint; but if we recollect that, after having calculated for one of the images the effect of interference corresponding to the difference \( d \) in the path, we are obliged, when we proceed to the other image, in order to obtain results conformable to experiment, to add \( \frac{1}{2}d \) to the difference of the paths, or to the effect of the difference of the velocities, this difficulty will disappear. Supposing \( d \) [nd] in the ordinary image to occasion the destruction of the red rays, \( d + \frac{1}{2}d \) [nd + \( \frac{1}{2}d \)] will correspond, on the contrary, to their most complete agreement in the extraordinary image, and these two images will exhibit tints rigorously complementary to each other, as the experiment shows. The colours developed by polarised light, in passing through crystallized plates being only, to speak correctly, portions of stripes produced by interference, we must expect to find in them, by varying the thicknesses of the plates, the same apparent deviations of the planes of polarisation as we found in the experiment on the narrow stripes produced by means of the rays transmitted by two plates of sulphate of lime, of which the principal sections were perpendicular to each other; and it has already been remarked that this analogy is supported by observation.
It can scarcely be doubted that this explanation will readily be adopted as fully satisfactory, by all those who will give themselves the trouble to examine it with sufficient attention; and the subject may now be concluded with the formulas which express the intensities of the ordinary and extraordinary images for the case of a polarised ray which has passed through one or two crystallized plates, with a perpendicular incidence, whatever may be the situation of their principal sections.
For a single plate, making unity the intensity of the primitive homogeneous pencil; \( i \) the angle made by the principal section of the plate with the primitive plane of polarisation; \( s \) the angle made by this same plane with the principal section of the rhomboid or of the achromatic prism, by means of which we analyse the emergent light; \( o-e \) the difference of the paths of the ordinary and extraordinary rays at their emergence; \( d \) the interval already explained; and \( e \) the circumference of the circle of which the diameter is unity: we shall then have,
for the ordinary image, \( \cos^2 s - \sin 2i \sin 2 (i-s) \sin^2 \frac{o-e}{d} \);
for the extraordinary image, \( \sin^2 s + \sin 2i \sin 2 (i-s) \sin^2 \frac{o-e}{d} \).
When the polarised pencil has passed through two plates, there must be an additional element in the formula, that is, the angle formed by the principal section of this second plate with the primitive plane of polarisation. Let this angle be \( t \), all the others retaining the same symbols, and let \( o'-e' \) be the difference of the paths belonging to the second plate; the intensity of the ordinary image afforded by a pencil of homogeneous light will then be represented by
\( \cos^2 s + \sin 2t \sin 2i \cos 2(t+i-s) \sin^2 \frac{o-e}{d} \)
\( \sin 2t \cos 2i \sin 2(t+i-s) \sin^2 \frac{o-e}{d} \)
\( \sin 2(t+i-s) \sin^2 \frac{o-e}{d} \)
\( \sin 2(t+i-s) \sin^2 \frac{o-e}{d} \);
and the intensity of the extraordinary image is found by subtracting this expression from unity.
If we calculate separately, according to these formulas, the intensities of the rays of the different colours which compose white light, we obtain the tint of the ordinary or of the extraordinary pencil, whether the light, previously polarised, has passed through one or two crystallized plates; and if, in this calculation, we employ for \( o-e \) and for \( o'-e' \) the values corresponding to the individual double refractions to which the different species of rays are subject, we shall find the most perfect agreement between the formulas and the experiments, even in those crystals which afford tints that appear to bear no resemblance to those of the coloured rings of Newton.
The phenomena of these thin plates, therefore, which seemed to some persons to afford an unexceptionable demonstration of the system of emission; which seemed to require the assistance of the most singular oscillations of the molecules of light; which seemed to enable them to discover, in those particles, an axis of rotation, a pair of poles, and an equator; and even a kind of precession of the equinoctial points: these phenomena are in fact only, as we have just seen, the immediate and unavoidable consequences of the simple but inexhaustible laws of interference.
**SECT. X.—ON CIRCULAR POLARISATION.**
The kind of polarisation which is now to be considered differs essentially from those which have been hitherto examined.
Suppose that, having polarised a pencil of light, we cause it to undergo twice over, at the angle of 54°, a total reflection within a parallelopiped of glass, as seen in the figure; the new planes of reflection being also supposed to be inclined at an angle of 45° to the plane of primitive polarisation; the emergent pencil will then have acquired some particular properties, which are very remarkable.
When this emergent pencil is analysed with a rhomboidal crystal, it is constantly decomposed into two rays of the same intensity, whatever may be the direction of the principal section. From this circumstance it might be supposed that it had resumed the character of ordinary light; but if it be transmitted through a crystallized plate before it is subjected to the action of the rhomboid, we shall soon perceive a distinction: for, in this case, common white light would afford two white images of the same intensity, while the light of the parallelopped is divided into two pencils, both strongly coloured.
This new kind of rays has also some other peculiarities. It has already been remarked (Sect. V.), that one or more total reflections make no difference in the properties of ordinary light; but they modify, on the contrary, in a remarkable manner, the pencil which has passed through the parallelopped: for this pencil resumes all the qualities of polarised light when it has been subjected to two total reflections similar to the first, whatever may be the azimuth of the latter planes of reflection with regard to the former.
The pencil of light in question is decomposed, then, into two coloured images, when it is only analysed by a rhomboid after having been transmitted through a crystallized plate; but it must be remarked, that the colour of each of these images, on the chromatic circle of Newton, is about one fourth of the circumference distant from the place which the colour of the same image would have occupied if we had employed only common polarised light.
It may also be added, as another distinguishing character, that this last kind of light gives rise to no phenomena of colour after being transmitted through plates of rock-crystal perpendicular to the axis, or through columns of oil of turpentine, or of other essential oils.
A polarised ray, modified by two total reflections, has therefore some very particular characters, which distinguish it equally from a direct ray and from an ordinary polarised ray; and as these characters have no relation to the different sides of the ray, the modification thus obtained has received the name of circular polarisation. If any person
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1 See the article Orries, p. 301. should think the denomination too little supported by the facts described, he may be informed, that it has been partly derived from some theoretical considerations, which appear to justify it.
The mode of obtaining the circular from the ordinary polarisation by means of two total reflections having been described, there remains to be mentioned a very remarkable method of procuring the circular polarisation more immediately from common light.
We cut out of a column of rock-crystal a very obtuse prism, with its two faces forming an angle of about 150°, and equally inclined to the axis of the column, and we make it achromatic as well as we can, with two prisms of glass cemented to its opposite faces: or, since this method must always be imperfect, we employ, instead of the glass, two other prisms of rock-crystal, taken from pieces which possess the opposite qualities which have been described in speaking of plagiedral crystals; and this arrangement has also the advantage of a separation of the images to a double distance. We obtain by means of this little combination the effects of double refraction; but the two pencils to which it gives birth, when they have been transmitted in a direction parallel to the axis of the column, have not acquired the same modification which Iceland spar, for example, would produce in them, but they are circularly polarised. Thus, if we analyse them with a rhomboid, they are always divided into two pencils of equal intensity; and when they have undergone two total reflections within a parallelopiped of glass at an angle of incidence of 54°, they are found completely polarised in two planes inclined 45° to the plane of reflection, the plane of polarisation of the one being to the right, and that of the other to the left.
There exists, therefore, a particular kind of double refraction, which communicates to the rays of light a circular polarisation, as the double refraction of the Iceland crystal produces in them the ordinary polarisation.
It follows, besides, from all the phenomena of this class, and from the general laws of interference, that a pencil circularly polarised may be considered as composed of two ordinary pencils polarised at right angles, one of them, however, being supposed to be retarded in its path, in comparison with the other, by a quarter of the interval which has been already noted by d.
The properties of rays circularly polarised afford us a very curious mode of reproducing exactly all the phenomena of colours, that we have observed in the plates of rock-crystal cut perpendicularly to the axis, and in certain liquids.
For this purpose we must place a thin crystallized plate between two parallelopipeds of ordinary glass crossed at right angles, and similar to that which has been already represented by a figure. A pencil, passing perpendicularly through the first parallelopiped, undergoes within it two total reflections at the angle of 54°; after its emersion, it is transmitted by the crystallized plate, and further on it enters the second parallelopiped, and is again twice reflected within it, but in a plane perpendicular to that of the former reflections: at last it emerges into the air, perpendicularly to the last surface of the parallelopiped. Now we may always obtain, in the emergent ray, the appearances of a polarised ray which passed through a plate of rock-crystal cut perpendicularly to the axis, or a certain depth of the essential oil of turpentine; and it will be sufficient for this purpose, that the pencil incident upon the first parallelopiped should be previously polarised, and that the axis of the interposed crystalline plate should make an angle of 45° with the two planes of the total reflections.
### Sect. XL—Enumeration of the Principal Minerals Which Are Possessed of Particular Properties with Respect to Polarisation. Added by the Translator.
i. All such regular crystals as belong to the rhomboïdal or pyramidal systems of crystallization, described by Mobs, are found to have one axis of double refraction, coincident with the crystallographic axis of these solids.—(Edinburgh Encyclopaedia, article Optics, p. 572.) The axis of a refraction governed by the law of the oblate spheroid is called by Sir David Brewster a negative axis, by M. Biot a regulative axis, and the axis of the oblong spheroid a positive or an attractive axis; the crystals are distinguished by the signs — and + respectively.
1. Rhomboid with obtuse summit. - Carbonate of lime. - Carbonate of lime and magnesia. - Rubellite. - Corundum. - Sapphire. - Emerald. - Beryl. - Phosphate of lime. - Zircon. - Oxide of tin. - Tungstate of lime. - Titanite. - Idocrase. - Wernicke. - Quartz. - Phosphate of lead.
2. Rhomboid with acute summit. - Ruby. - Cinnabar. - Nepheline, or Sommitte. - Arseniate of copper. - Arseniate of lead. - Mellite. - Molybdate of lead. - Octohedrite. - Melonite. - Uranite.
3. Regular hexagonal prism. - Phosphato-arsenate of lead.
ii. Besides the crystals belonging to these classes, there are others of which the primitive form has not been determined, but in which the position of the single axis of double refraction has been ascertained.
| Crystals | Axis of Double Refraction | |---------|--------------------------| | Mica from Karat. | Perpendicular to the laminae. | | Mica with amanthus. | Perpendicular to the laminae. | | Hydrot of strontia. | Perpendicular to the quadrangular plate. | | Arseniate of potass. | Axis of quadrangular prism. | | Muriate of lime. | Axis of hexagonal prism. | | Nitrate of strontia. | Axis of hexagonal prism. | | Nitrate of soda. | Axis of obtuse rhomb. | | Subphosphate of potass. | Axis of quadrangular prism. | | Sulphate of nickel and copper. | Axis of quadrangular prism. | | Hyposulphate of lime (Herbert). | Axis of hexagonal tables. | | Boracite. | Axis of cubical rhomb. | | Apophyllite from Uton. | Axis of prism. | | Apophyllite surcomposita. | Perpendicular to the plate. | | Sulphate of potass and iron. | Axis of hexagonal prism. | | Sulfate of copper and lime. | Axis of prism. | | Hydrate of magnesia. | Perpendicular to the laminae. | | Ice. | Axis of hexagonal prism or rhomb. |
iii. The crystals which have been enumerated appear to be symmetrical with respect to a single axis, so as to have similar properties in every plane passing through that axis; but Sir David Brewster discovered, in 1816, that in a great multitude of crystals the refractive powers are different in... different planes passing through the principal axis; the phenomena and the laws of refraction being such as may be explained from the combined effects of two axes perpendicular to each other, and unequal in the ellipticity of the spheroids to which they belong.
All regular crystals which belong to the primitive system of Mohs, or whose primitive forms are the right prism, with its base a rectangle, a rhomb, or an oblique parallelogram; the oblique prism, with its base a rectangle, a rhomb, or an oblique parallelogram; or the rectangular and rhomboïdal octaedron have two axes of double refraction, coincident with some permanent line in the primitive form.
1. Thus, all the combinations of the sulphuric, tartaric, and acetic acids, with single earthy, metallic, and alkaline bases, have two axes of double refraction.
2. The two apparent axes of crystals with two axes [or the lines in which the double refraction seems to be neutralised] have no symmetrical relation either with the faces or prominent lines of the primitive or secondary forms of minerals.
3. The two rectangular axes, the principal one of which is in the same plane with the two apparent axes, and equidistant from them, have a constant symmetrical relation to the faces and axes of the primitive forms in which they crystallize.
4. The line exhibited at any point of the sphere, by the joint properties of two axes, is represented by the diagonal of a parallelogram, of which the sides correspond to the tints belonging to the separate axes, and make with each other an angle twice as great as the angle formed by the planes passing through the given point and the two axes; and the increment or decrement of the square of the velocity of the light may be computed in the same manner as this line representing the tint.
iv. The necessity of any very minute investigation of the properties of particular substances, with regard to double refraction, is in great measure superseded by the very important facts, first observed by Sir David Brewster in 1814 and 1815, relating to the effects of heat and compression in producing double refraction and polarisation. He found that compression was capable of producing colour when a soft animal jelly was only touched by the finger; and that when a "negative" crystal, like that of the carbonate of lime, is compressed in the direction of the axis, the tints that it affords "descend," and that they "rise" when it is dilated: whence it seems to follow, that simple dilatation of a homogeneous substance in a given line will constitute that line the axis of an oblate spheroid; but the results require to be distinguished by a greater variety of experiments. M. Fresnel has succeeded in exhibiting not only colours by a strong pressure, but a very manifest reduplication of the image of a line, seen through a piece of glass strongly compressed by screws; and he performed this experiment very successfully at a meeting of the Parisian Academy of Sciences in 1822.
SECT. XII.—HISTORICAL DETAILS RESPECTING THE DISCOVERY OF THE DIFFERENT PROPERTIES OF LIGHT CONCERNED IN POLARISATION.
It will now be necessary to enumerate the natural philosophers to whom we are indebted for the discoveries of which the importance has been explained in this article; and it will be convenient to do this in the order of the sections.
i. Huygens appears to be the first person that observed, in the two pencils derived from a single one, by means of double refraction, the existence of particular properties after their passage through the crystal, which they did not possess when they entered it. "It seems," he says, "that we are obliged to conclude, that the undulations of light, by passing through the first crystal of Iceland spar, acquire a certain form, or a certain disposition, by which, when they fall on the substance of a second crystal, in a certain position, they are enabled to affect both the kinds of matter which serve for the two species of refraction; and when they fall on it in another position, they only act on one of these substances."
Thus, according to this great philosopher, in the act of double refraction, the undulation, or the ray, changes its form, and loses its symmetry, so as to give room for the distinction of its different sides, or, changing the expression only, its different poles.
Huygens is therefore the first that observed a phenomenon of polarisation, and this discovery was made in 1678, though it was not published till 1690. From the time of Huygens to the year 1809, no observer, with the exception of the immortal author of the Opticks, had studied the subject of double refraction in this point of view; and it must even be acknowledged that, with respect to the facts in question, nothing was added by Newton to what the Dutch mathematician had discovered. He only insisted much more strongly on the necessity of admitting poles in each of the rays derived from the subdivision which light undergoes in passing through an Iceland crystal. To Malus belongs the honour of having brought back the attention of natural philosophers to the properties of light which form the subject of this article. It was he that first pointed out the singular phenomena exhibited by the ordinary and extraordinary rays when they meet with transparent reflecting surfaces at certain inclinations; and it is to him that we are indebted for the mathematical law which appears to connect intensities of the different pencils into which the light is divided when it passes through two rhomboids of the spar in succession.
ii. The anecdote, which has been often told, of the fortunate circumstance that led Malus to the discovery of the polarisation of light by reflection from transparent substances, is perfectly correct. This philosopher, so early arrested in his pursuit of science by a premature death, and so universally lamented by his friends, has often related to the author of this article, that as he happened to be decomposing, by means of a rhomboid of carbonate of lime, towards the end of the year 1808, the light of the setting sun, reflected from the glass of the windows of the Luxembourg, he first observed the difference of the intensity of the two images in different positions of the rhomboid; but it is not true, as has sometimes been stated, that one of the images actually disappeared in this observation; for the polarisation of the light, at the moment of the experiment, was only partial. A similar difference of intensity must have been before the eyes of mineralogists whenever they had been trying to examine the double refraction of crystals, and had projected the needle which they used for an object of view on a clear and serene sky, which affords a curtain of polarised light. But the fact had not excited their attention. Malus observed it, was struck with all its importance, completely analysed it in all its forms, with the most singular sagacity, and thus became the creator of a new branch of optics. It is to this celebrated observer that we are indebted for all the experiments related in the second section.
iii. The measurements related in the third section, which appear to show that, at equal angular distances above and below the angle of complete polarisation, the reflected rays contain nearly equal quantities of polarised light, were obtained by M. Arago.
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1 Mémoires d'Académie, ii. 8vo, Par., 1809. 2 Ibid. ii. iv. It was Sir David Brewster that discovered the remarkable law which connects the angle of complete polarisation with the refractive density of the substance. It was communicated to the Royal Society on the 16th March 1815.
The relation of the angle of complete polarisation, at the second surface of a transparent medium, to the angle belonging to the first surface, had before been shown by Malus. The same relation may be extended to the angles, at the same surfaces, which afford polarisation of equal proportions of the light falling on them. So that the rule given by Malus is only a particular case of a general theorem, which M. Arago has deduced from a long series of experiments that have not been published.
v. It was discovered also by Malus, that the pencil transmitted by a transparent plate is partially polarised in a plane at right angles to the plane of polarisation of the reflected pencil. This fact was communicated to the Institute on the 11th March 1811, and published the next day in the Moniteur. See also the Memoirs of the Mathematical Class for 1810. The photometrical experiments of M. Arago have established a relation between these two kinds of polarisation which had been unobserved by Malus; it is contained in this simple enunciation. "The quantity of polarised light contained in the pencil which any transparent plate transmits, is exactly equal to the quantity polarised in the contrary direction, which is found in the light reflected by the same plate." These experiments were made in 1812, but they were first published in 1814 by M. Biot, to whom M. Arago had communicated them.
It follows from this law, as M. Arago had also observed, that at the angle of total reflection, and at all greater obliquities, the light wholly escapes polarisation.
The phenomena exhibited by piles of plates were analysed by Malus immediately after his discovery of polarisation by refraction.
The observation that some natural bodies, agates, for example, act on light precisely like these piles of plates, is due to Sir David Brewster.
vi. The laws and experiments related in the sixth section belong to M. Fresnel. Malus had before attempted to discover in what manner the planes of polarisation changed their directions; but there are several inaccuracies in the results which he has published. The formulae of M. Fresnel are some of the most valuable additions that have been made of late to the science of optics. A general account of the deviations undergone, in reflection, by the plane of polarisation of a ray previously polarised, is found in two memoirs presented to the academy by M. Fresnel, on the 24th November 1817, and the beginning of January 1818; but the mathematical laws of the phenomena were not discovered and published till 1821.
vii. M. Arago and M. Fresnel are hitherto the only philosophers who have examined the effect of polarisation in modifying the phenomena of interference. The memoir in which they first recorded the results which are inserted in the seventh section, appeared in the Annales de Chimie for 1819, vol. x.
viii. It was M. Arago that first observed the changes in the properties of polarised rays when they pass through crystalline plates. He showed that they acquire the property of being divided by calcareous spar into two coloured pencils, and of being reflected by transparent substances with tints which vary according to the angular position of their surfaces. His memoir was read to the Institute on the 11th of August 1811, and printed in the volume of the Mémoires of that year.
Sir David Brewster published some similar experiments in his Treatise on Philosophical Instruments, which appeared in 1813.
The phenomena exhibited by plates cut perpendicularly to the axis were also described by M. Arago in the same memoir.
We are indebted to M. Biot for the rule, according to which the deviation of the poles is effected, whether on account of the particular kind of "each luminar molecule," or on account of the more or less considerable thicknesses of the plates by which these "molecules" have been transmitted. His memoir was read to the Institute in September 1818, and printed shortly afterwards.
Mr Herschel is the author of the curious observation relating to the facets of plagiedral crystals.
The extension of the properties of plates perpendicular to the axis, by which they are assigned to the strata of certain liquids, was made by M. Biot in 1815.
ix. The laws of the depolarisation produced by crystalized plates, parallel to the axis, are reduced to the three following.
1. The motion of the plate in its plane does not alter the tints of the images furnished by a rhomboid. This result is implicitly comprehended in the first memoir of M. Arago; since in the description of all the motions which cause a change of the tints, the motion of the plate in its own plane is not included.
2. The tints of the two images are those of the coloured rings of Newton, seen by reflection or by transmission. This law had been laid down by M. Biot; but Mr Herschel has shown in the Cambridge Transactions that it is not universally true, so that its theoretical importance is lost.
3. In a crystal of variable thickness, the same phenomena of polarisation are reproduced at thicknesses which form a series like that of the coloured rings of Newton. When we examine with a rhomboid a crystal cut in a proper manner, so that two of its faces form an angle, and which is projected on a background affording polarised light, each image appears bordered by regular streaks, parallel to the angular edge of the prism, and separated by equal spaces. M. Arago, when he described this phenomenon in August 1811, advanced it as a sufficient proof of this third law. The Academy of Sciences, and M. Laplace in particular, did not admit the conclusion as demonstrated, and a direct admeasurement of the thicknesses was required. Count Rumford, who was present, offered the use of an instrument which he had employed for some other purposes, and which seemed to promise a sufficient degree of precision; the common comparer was also mentioned as proper for the purpose. M. Arago, upon these suggestions, undertook to make these further experiments. But M. Biot anticipated him; and he has therefore a claim to the first direct demonstration of the law of the thicknesses.
4. The brilliant tints of each of these images may be calculated from the laws of interference, according to the difference of the paths and of the velocities of the ordinary and extraordinary rays. This important remark is due to Dr Thomas Young, who is believed to have published it in the Quarterly Review, xi. 1814, p. 42-49.
It affords the true key of all these phenomena. It must
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1 Philosophical Transactions, 1815. 2 Mem. d'Arcueil, ii. 3 See Recherches Expérimentales et Mathématiques sur les Mouvements des Molécules de Lumière, 1814. 4 Treatise on New Philosophical Instruments, 8vo, Edinburgh, 1813. 5 Mem. Inst. 1810. 6 See Annales de Chimie et de Physique, xvii. 7 Transactions of the Philosophical Society of Cambridge, i.; see also the article Optics, p. 502. 8 Bulletin des Sciences, Dec. 1815; see also the article Optics, p. 504. 9 Phil. Trans. 1818, p. 243, 252; see also the article Optics, p. 487. Polarisation of Light.
However be added, that Dr Young had not explained either in what circumstances the interference of the rays can take place, nor why we see no colours unless the crystallized plates are exposed to light previously polarised. The new properties which required to be combined with the ordinary laws of interference, in order to obtain a complete explanation of the facts, were discovered by M. Arago and M. Fresnel, as they have been described in the seventh section. The dates of the memoirs in which M. Fresnel published this explanation are 1816 and 1818.
The ingenious and delicate experiments which have been employed as tests of the inaccuracy of the theory of moveable polarisation are also due to M. Fresnel, one of the most inventive theorists as well as one of the most skilful experimenters that have devoted themselves to science in this or in any age: the formulae found at the end of the section are also M. Fresnel's.
x. The phenomena of circular polarisation were discovered by M. Fresnel, who described and analysed them in a memoir read to the Academy of Sciences in November 1817 and in January 1818. His later researches on this subject are inserted in the Bulletin of the Philomathic Society, December 1822 and February 1823.
There are a few other particulars, relating to the development of tints by repeated reflections at the surfaces of metallic mirrors, to the rings seen in crystallized plates cut perpendicularly to the axis when examined with a rhomboid held very near them, and to the phenomena of the absorption of polarised light by certain crystals. The first have hitherto been reduced to no direct analogy with other affections of polarised light; the second seem to be derived from a modification of the laws already explained in the ninth section; and the discovery of the scattered phenomena belonging to the third head has been disputed by M. Biot and Sir David Brewster, but they appear to have been first observed by M. Arago in 1814.
[SECT. XIII.—THEORETICAL INVESTIGATIONS INTENDED TO ILLUSTRATE THE PHENOMENA OF POLARISATION. ADDED BY THE TRANSLATOR.]
We are led, from the facts which have been enumerated in the eleventh section, to the remarkable coincidence between the discoveries of Sir David Brewster respecting crystals with two axes, and a theory which had been published a few years earlier in order to illustrate the propagation of an undulation in a medium compressed or dilated in a given direction only, and to prove that such an undulation must necessarily assume a spheroidal form, upon the mechanical principles of the Huygenian theory. As every contribution to the investigation of so difficult a subject may chance to be of some value, it will be worth while to copy this demonstration here, from the Quarterly Review for November 1809, vol. ii. p. 345.
"The proposition to be demonstrated was this: An impulse is propagated through every perpendicular section of a lamellar elastic substance in the form of an elliptic undulation.
"When a particle of the elastic medium is displaced in an oblique direction, the resistance produced by the compression is the joint result of the forces arising from the elasticity in the direction of the lamina, and in a transverse direction; and if the elasticities in these two directions were equal, the joint result would remain proportional to the displacement of the particle, being expressed, as well in magnitude as in direction, by the diagonal of the parallelogram, of which the sides measure the relative displacements, reduced to their proper directions, and express the forces which are proportional to them. But when the elasticity is less in one direction than in the other, the corresponding side of the parallelogram expressing the forces must be diminished, in the ratio which we may call that of \( \frac{1}{m} \); and the diagonal of the parallelogram will no longer coincide in direction with the line of actual displacement, so that the particle displaced will also produce a lateral pressure on the neighbouring particle of the medium, and will itself be urged by a lateral force. This force will, however, have no effect in promoting the direct propagation of the undulation, being probably employed in gradually changing the direction of the actual motions of the successive particles; and the only efficient force of elasticity will be that which acts in the direction in which the undulation is advancing, and which is expressed by the portion of the line of displacement, cut off by a perpendicular falling on it from the end of the diagonal of the parallelogram of forces; and the comparative elasticity will be measured by this portion, divided by the whole line of displacement. Calling the tangent of the angle formed by the line of displacement with the line of greatest elasticity \( t \), the radius being 1, the force in this line being also 1, the transverse force will be expressed by \( nt \), the line of displacement by \( \sqrt{(1 + u)} \), its diminution by \( \frac{(1 - m)u}{\sqrt{(1 + u)}} \), the diminished portion, which measures the force, by \( \sqrt{(1 + u)} \cdot \frac{(1 - m)u}{\sqrt{(1 + u)}} \), and the elasticity, in the given direction, by \( \frac{1 + mtt}{1 + u} \).
Hence it follows, that the velocity of an impulse, moving in that direction, will be expressed by \( \sqrt{\frac{1 + mtt}{1 + u}} \).
"It is next to be proved, that the velocity of an elliptical undulation, increasing so as to remain always similar, by means of an impulse propagated always in a direction perpendicular to the circumference, is such as would take place in a medium thus constituted. It is obvious that the increment of each of the diameters of the increasing figure must be proportional to the whole diameter; and this increment, reduced to a direction perpendicular to the curve, will be proportional to the perpendicular falling on the conjugate diameter, which will measure the velocity. We are therefore to find an expression for this perpendicular when it forms an angle with the greater axis, of which the tangent is \( t \). Let the greater semiaxis be 1, and the smaller \( n \); then the tangent of the angle formed with the greater axis by the conjugate diameter being \( \frac{1}{t} \), the tangent of the angle subtended by the corresponding ordinate of the circumscribing circle is found \( \frac{1}{nt} \); and the semidiameter itself, equal to unity, reduced in the ratio of the secants of these angles, that is, to \( n \sqrt{\frac{1 + u}{1 + mtt}} \); but, by the known property of the ellipse, the perpendicular required is equal to the product of the semiaxis, divided by this semidiameter, that is, to \( \sqrt{\frac{1 + mtt}{1 + u}} \); we have therefore only to make \( nu = m \), and the velocity in the given medium will always be such as is required for the propagation of an undulation, preserving the form of similar and concentric spheroids, of which the given ellipse represents any principal section.
"If the whole of the undulation were of equal force, this
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1 See the article Optics, p. 505. 2 See Biot's Recherches sur les Mouvements des Molécules de Lumière, 4to, Paris, 1814; p. lxxxvi. reasoning would be sufficient for determining its motion; but when one part of it is stronger than another, this superiority of pressure and motion will obviously be propagated in the direction of the actual resistance produced by the displacement of the particles, since it is this resistance which carries on the pressure, and consequently propagates the motion. It is very remarkable that the direction of the resistance will be found, on the supposition which has been advanced respecting the constitution of the medium, to coincide everywhere with the diameters of the ellipse when the displacement is perpendicular to the surface.
For it is proved by authors on conic sections, that the subnormal of the ellipse is to the absciss as the square of the lesser axis is to the square of the greater, that is, in this case, as \( m \) to 1, or as \( m \) to 1; but if we divide the ordinate to the same ratio of \( m \) to 1, and join the point of division with the extremity of the subnormal, this line, which will evidently be parallel to the diameter, will express, as we have already seen, the direction of the force when the normal represents that of the displacement. An immediate displacement in the direction of any diameter, making an angle with the axis of which the tangent is \( t \), would give a velocity of \( \sqrt{\frac{1 + mt}{1 + u}} \), while the increment of the diameter would require a velocity of \( \sqrt{\frac{m + mt}{m + u}} \), which does not vary in the same proportion. It must, however, be remembered, that the rectilinear direction of the beam is not supposed to depend on this circumstance alone. Huygens considers each point of the surface of the crystal, on which a beam of light impinges, as the centre of a new undulation, which spreads in some measure in every direction, but produces no perceptible effect except where it is supported by, and co-operates with, the neighbouring undulations, that is, in the surface which is a common tangent of the collateral undulations; but if this principle were applied to extraordinary refraction without the assistance of the obliquity of force, which may be deduced from the supposition of a stratified medium, it would lead us to expect that the elementary impulses, being propagated in a curvilinear trajectory, might be intercepted by an object not situated in the rectilinear path of the beam; a conclusion which is not warranted by experiment.
However satisfactorily this mode of viewing the extraordinary refraction may be applied to the subsequent discoveries relating to the effects of heat and compression, there is another train of ideas, which arises more immediately from the phenomena of polarisation, and which might lead us to a more distinct notion of the separation of the pencil into two or more portions, though it does not seem to comprehend so entirely the phenomena depending on spheroidal undulations.
We may begin this mode of considering the subject in the words which have already been employed in the article Chromatics, vol. vi. p. 649. "If we assume as a mathematical postulate, in the undulatory theory, without attempting to demonstrate its physical foundation, that a transverse motion may be propagated in a direct line, we may derive from this assumption a tolerable illustration of the subdivision of polarised light by reflection in an oblique plane. Supposing polarisation to depend on a transverse motion in the given plane; when a ray completely polarised is subjected to simple reflection in a different plane, at a surface which is destitute of any polarising action, and which may be said to afford a neutral reflection, the polar motion may be conceived to be reflected, as any other motion would be reflected, at a perfectly smooth surface, the new plane of the motion being always the image of the former plane; and the effect of refraction will be nearly of a similar nature. But when the surface exhibits a new polarising influence, and the beams of light are divided by it into two portions, the intensity of each may be calculated, by supposing the polar motion to be resolved instead of being reflected, the simple velocities of the two portions being as the cosines of the angles formed by the new planes of motion with the old, and the energies, which are the true measure of the intensity, as the squares of the sines. We are thus insensibly led to confound the intensity of the supposed polar motion with that of the reflected light itself; since it was observed by Malus, that the relative intensity of the two portions into which light is divided under such circumstances, is indicated by the proportion of the squares of the cosine and sine of the inclination of the planes of polarisation. The imaginary transverse motion might also necessarily be alternate; partly from the nature of a continuous medium; and partly from the observed fact, that there is no distinction between the polarisations produced by causes precisely opposed to each other in the same plane." Another analogous hint is found in the Philosophical Transactions for 1818, p. 273. "Supposing the experiments to be perfectly represented by [Dr Brewster's] general law, it will follow that the tint exhibited depends not on the difference of refractive densities in the direction of the ray transmitted, but on the greatest difference of refractive densities in directions perpendicular to that of the ray. These two conditions lead to the same result where the effect of one axis only is considered, but they vary materially where two axes are supposed to be combined. There can be little doubt that the direction of the polarisation, in such cases, must be determined by that of the greatest and least of the refractive densities in question?" A "supposition" which Dr Brewster finds "quite correct."
We may add again to these hints the consideration, that when simple pressure or extension in the direction of any given axis produces a spheroidal undulation in a medium before homogeneous, this state is always accompanied by the condition, that a ray describing the axis, while the densities in all transverse directions remain equal, undergoes no subdivision; but that a ray moving in the plane of the equator, to which the perpendiculars are the axis and another equatorial diameter, undergoes the greatest possible separation into parts that are respectively polarised in the planes passing through these directions.
From these phenomena we are led to be strongly impressed with the analogy of the properties of sound, as investigated cursorily by Mr Wheatstone, and in a more elaborate manner by the multiplied experiments of M. Savart, which have shown that, in many cases, the elementary motions of the substances transmitting sound are transverse to the direction in which the sound is propagated, and that they remain in general parallel to the original impulse.
The next transition carries us from the mathematical postulate here mentioned, to the physical condition assumed by M. Fresnel, that the relative situation of the particles of the etherial medium with respect to each other, is such as to produce an elastic force tending to bring back a line of particles which has been displaced, towards its original situation, by the resistance of the particles surrounding the line; and at the same time to impel these particles in its own direction, and in that direction only, or principally, while the aggregate effect is propagated in concentric surfaces.
This hypothesis of M. Fresnel is at least very ingenious, and may lead us to some satisfactory computations; but it is attended by one circumstance which is perfectly appalling in its consequences. The substances on which M. Savart made his experiments were solids only, and it is only to solids that such a lateral resistance has ever been attributed; so that if we adopted the distinctions laid down by the reviver of the undulatory system himself in his Lectures, it might be inferred that the luminiferous ether per- vading all space, and penetrating almost all substances, is not only highly elastic, but absolutely solid. The passage in question is this:
"The immediate cause of solidity, as distinguished from liquidity, is the lateral adhesion of the particles to each other, to which the degree of hardness or solidity is always proportional. This adhesion prevents any change of the relative situation of the particles, so that they cannot be withdrawn from their places, without experiencing a considerable resistance from the force of cohesion, while those of liquids may remain equally in contact with the neighbouring particles, notwithstanding their change of form.
When a perfect solid is extended or compressed, the particles, being retained in their situations by the force of lateral adhesion, can only approach directly to each other, or be withdrawn farther from each other; and the resistance is nearly the same as if the same substance, in a fluid state, were enclosed in an unalterable vessel, and forcibly compressed or dilated. Thus the resistance of ice to extension or compression is found by experiment to differ very little from that of water contained in a vessel; and the same effect may be produced even when the solidity is not the most perfect that the substance admits; for the immediate resistance of iron or steel to flexure is the same, whether it may be harder or softer. It often happens, however, that the magnitude of the lateral adhesion is so much limited as to allow a capability of extension or compression, and it may yet retain a power of restoring the bodies to their original form by its re-action. This force may even be the principal or the only source of the body's elasticity. Thus, when a piece of elastic gum is extended, the mean distance of the particles is not materially increased...and the change of form is rather to be attributed to a displacement of the particles than to their separation to a greater distance from each other, and the resistance must be derived from the lateral adhesion only. Some other substances, also, approaching more nearly to the nature of liquids, may be extended to many times their original length, with a resistance continually increasing; and in such cases there can scarcely be any material changes of the specific gravity of these substances. Professor Robison has mentioned the juice of bryony as affording a remarkable instance of such viscosity.
"It is probable that the immediate cause of the lateral adhesion of solids is a symmetrical arrangement of their constituent parts. It is certain that almost all bodies are disposed, in becoming solid, to assume the form of crystals, which evidently indicates the existence of such an arrangement; and all the hardest bodies in nature are of a crystalline form. It appears, therefore, consistent both with reason and with experience to suppose, that a crystallization more or less perfect is the universal cause of solidity. We may imagine, that when the particles of matter are disposed without any order, they can afford no strong resistance to a motion in any direction; but when they are regularly placed in certain situations with respect to each other, any change of form must displace them in such a manner as to increase the distance of a whole rank at once; and hence they may be enabled to co-operate in resisting such a change. Any inequality of tension in a particular part of a solid is also probably so far the cause of hardness, as it tends to increase the strength of union of any part of a series of particles which must be displaced by a change of form."
It must, however, be admitted, that this passage by no means contains a demonstration of the total incapability of fluids to transmit any impressions by lateral adhesion, and the hypothesis remains completely open for discussion, notwithstanding the apparent difficulties attending it; which have appeared to bring us very near to the case stated in the same lectures as a possible one, that there may be independent worlds, some existing in different parts of space, others pervading each other unseen and unknown in the same space. We may perhaps accommodate the hypothesis of M. Fresnel to the phenomena of the ordinary and extraordinary refraction, by considering the undulations as propagated through the given medium in two different ways; some by the divergence of the elementary motions in the direction of the ray, and others by their remaining parallel to the direction of the impulse or of the polarisation. The former must be supposed to furnish the spheroidal, the latter the spherical refraction. It would indeed follow that the velocity of the spherical undulation ought to vary by innumerable degrees, within certain limits, according to the direction of the supposed elementary motion; whilst in fact the actual velocity of the spherical undulations seems to be uniformly equal to the velocity in the direction of the axis; but this objection may be obviated by supposing the surface so constituted, that for some unknown reason the parallel elementary motion can only be propagated in the regular manner when it takes place in the direction of the axis, or when it is made to assume that direction; a condition not very simple or natural, but by no means inconceivable, unless we saw any reason to consider the adhesion as a constant force, independent of the direction, and equal to the least or greatest elasticity, or unless it were possible to derive the phenomena of two supposed axes of polarisation, which M. Fresnel has explained on the hypothesis of two spheroids, from the supposition of two spherical undulations propagating oblique elementary motions in the direction of the actual polarisation, as already determined for these crystals.
If these conjectures should be found to afford a single step in an investigation so transcendently delicate, it will be best to pause on them for a time, and to wait for further aid from a new supply of experiments and observations.
(T. T. T.)
POL DE LEON, a city of France, in the department of Finisterre, and arrondissement of Morlaix. It is situated on a hill near the sea, about a mile from its haven Penpont, which is accessible to small vessels. It is the see of a bishop, and has a cathedral and other churches, 1005 houses, and 6150 inhabitants, who make linen goods, and some pottery ware. Long. 3. 58. 26. W. Lat. 48. 40. 55. N.